UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                    -----------------------------------------


                                  SCHEDULE 13G
                    Under the Securities Exchange Act of 1934
                    -----------------------------------------


                                   ENZON, INC.
                    -----------------------------------------
                                (Name of Issuer)


                     Common Stock, par value $0.01 per share
                   ------------------------------------------
                         (Title of Class of Securities)

                                    293904108
                                  -------------
                                 (CUSIP Number)

                                  July 14, 1998
                                -----------------
             (Date of Event Which Requires Filing of the Statement)

         Check the  appropriate box to designate the rule pursuant to which this
Schedule is filed:

                                            [ ]   Rule 13d-1(b)
                                            [x]   Rule 13d-1(c)
                                            [ ]   Rule 13d-1(d)

         *The  remainder  of this cover page shall be filled out for a reporting
person's  initial  filing on this  form with  respect  to the  subject  class of
securities,  and for any subsequent amendment containing information which would
alter disclosures provided in a prior cover page.

         The information  required in the remainder of this cover page shall not
be deemed to be "filed" for the purpose of Section 18 of the Securities Exchange
Act of 1934 ("Act") or otherwise  subject to the  liabilities of that section of
the Act but shall be subject to all other  provisions of the Act  (however,  see
the Notes to Schedule 13G).

                                        1




CUSIP NO. 293904108                   13G                           page 2 of 12
- -------------------                   ---                           ------------



    1     NAMES OF REPORTING PERSONS
          S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

          Paramount Capital Asset Management, Inc.

    2     CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP        (a)|_| (b)|_|

    3     SEC USE ONLY

    4     CITIZENSHIP OR PLACE OF ORGANIZATION
          Delaware

                                               5      SOLE VOTING POWER
                 NUMBER OF                            None
                  SHARES                      
               BENEFICIALLY                    6      SHARED VOTING POWER
                 OWNED BY                             1,888,832
                   EACH
                 REPORTING                     7      SOLE DISPOSITIVE POWER
                  PERSON                              None
                   WITH
                                               8      SHARED DISPOSITIVE POWER
                                                      1,888,832
                                              

     9     AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
           1,888,832

    10     CHECK BOX IF THE AGGREGATE AMOUNT IN ROW 9 EXCLUDES CERTAIN SHARES*
                                                                             |_|
    11     PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)
           5.3%

    12     TYPE OF REPORTING PERSON*
           CO






CUSIP NO. 293904108                   13G                           page 3 of 12
- -------------------                   ---                           ------------



    1     NAMES OF REPORTING PERSONS
          S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

          Aries Domestic Fund, L.P.

    2     CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP        (a)|_| (b)|_|

    3     SEC USE ONLY

    4     CITIZENSHIP OR PLACE OF ORGANIZATION
          Delaware

                                               5      SOLE VOTING POWER
                 NUMBER OF                            None
                  SHARES                      
               BENEFICIALLY                    6      SHARED VOTING POWER
                 OWNED BY                             547,964
                   EACH
                 REPORTING                     7      SOLE DISPOSITIVE POWER
                  PERSON                              None
                   WITH
                                               8      SHARED DISPOSITIVE POWER
                                                      547,964
                                              

     9     AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
           547,964

    10     CHECK BOX IF THE AGGREGATE AMOUNT IN ROW 9 EXCLUDES CERTAIN SHARES*
                                                                             |_|
    11     PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)
           1.5%

    12     TYPE OF REPORTING PERSON*
           PN







CUSIP NO. 293904108                   13G                           page 4 of 12
- -------------------                   ---                           ------------



    1     NAMES OF REPORTING PERSONS
          S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

          The Aries Trust

    2     CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP        (a)|_| (b)|_|

    3     SEC USE ONLY

    4     CITIZENSHIP OR PLACE OF ORGANIZATION
          Cayman Islands

                                               5      SOLE VOTING POWER
                 NUMBER OF                            None
                  SHARES                      
               BENEFICIALLY                    6      SHARED VOTING POWER
                 OWNED BY                             1,340,868
                   EACH
                 REPORTING                     7      SOLE DISPOSITIVE POWER
                  PERSON                              None
                   WITH
                                               8      SHARED DISPOSITIVE POWER
                                                      1,340,868
                                              

     9     AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
           1,340,868

    10     CHECK BOX IF THE AGGREGATE AMOUNT IN ROW 9 EXCLUDES CERTAIN SHARES*
                                                                             |_|
    11     PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)
           3.8%

    12     TYPE OF REPORTING PERSON*
           OO (see Item 2)







CUSIP NO. 293904108                   13G                           page 5 of 12
- -------------------                   ---                           ------------



    1     NAMES OF REPORTING PERSONS
          S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

          Lindsay A. Rosenwald, M.D.

    2     CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP        (a)|_| (b)|_|

    3     SEC USE ONLY

    4     CITIZENSHIP OR PLACE OF ORGANIZATION
          United States

                                               5      SOLE VOTING POWER
                 NUMBER OF                            None
                  SHARES                      
               BENEFICIALLY                    6      SHARED VOTING POWER
                 OWNED BY                             1,888,832
                   EACH
                 REPORTING                     7      SOLE DISPOSITIVE POWER
                  PERSON                              None
                   WITH
                                               8      SHARED DISPOSITIVE POWER
                                                      1,888,832
                                              

     9     AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
           1,888,832

    10     CHECK BOX IF THE AGGREGATE AMOUNT IN ROW 9 EXCLUDES CERTAIN SHARES*
                                                                             |_|
    11     PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)
           5.3%

    12     TYPE OF REPORTING PERSON*
           IN




Item 1.

     (a)  Name of Issuer:

                           Enzon, Inc.

     (b)  Address of Issuer's Principal Executive Offices:

          Enzon, Inc.
          20 Kingsbridge Rd.
          Piscataway, NJ 08854-3969
          732-980-5911

Item 2.

     (a)  Name of Person Filing:

          This  statement  is  filed  on  behalf  of  Paramount   Capital  Asset
          Management,  Inc.  ("Paramount  Capital"),  Aries Domestic Fund,  L.P.
          ("Aries Domestic"), The Aries Fund, a Class of the Aries Trust ("Aries
          Trust")  and  Lindsay  A.   Rosenwald,   M.D.  ("Dr.   Rosenwald"  and
          collectively, "Reporting Parties"). See attached Exhibit A, which is a
          copy of their agreement in writing to file this statement on behalf of
          each of them.

          Dr.  Rosenwald  is an  investment  banker,  venture  capitalist,  fund
          manager and sole  shareholder  of  Paramount  Capital,  a Subchapter S
          corporation incorporated in Delaware. Paramount Capital is the General
          Partner  of Aries  Domestic,  a limited  partnership  incorporated  in
          Delaware,  and is the  Investment  Manager  to Aries  Trust,  a Cayman
          Islands Trust.

     (b)  Address of Principal Business Office or, if None, Residence:

          Paramount  Capital's,  Aries Domestic's and Dr.  Rosenwald's  business
          address is 787 Seventh Avenue,  48th Floor, New York, New York, 10019.
          The  business  address  for Aries  Trust is c/o  MeesPierson  (Cayman)
          Limited,  P.O. Box 2003,  British American Centre,  Phase 3, Dr. Roy's
          Drive, George Town, Grand Cayman.

     (c)  Citizen:

          Dr. Rosenwald is a citizen of the United States.

     (d)  Title of Class of Securities:

          Common Stock, $0.01 par value ("shares").

     (e)  CUSIP#:

          293904108

Item 3.   Check the box if this statement is filed pursuant to Rule 13d-1(c) [x]


                                        6




Item 4.   Ownership:

          For  information  concerning  the  ownership  of  Common  Stock of the
          Company by the Reporting Persons,  see Items 5 through 9 and 11 of the
          cover pages to this schedule 13G and footnotes thereto.

Item 5.   Ownership of Five Percent or Less of a Class:

          Not applicable.

Item 6.   Ownership of More than Five Percent on Behalf of Another Person:

          Not applicable.

Item 7.   Identification and Classification of the Subsidiary Which Acquired the
          Security Being Reported On By the Parent Holding Company:

          Not applicable.

Item 8.   Identification and Classification of Members of the Group:

          Not applicable.

Item 9.   Notice of Dissolution of Group:

          Not applicable.

Item 10.  Certification:

          By  signing  below I certify  that,  to the best of my  knowledge  and
          belief, the securities referred to above were not acquired and are not
          held for the purpose of or with the effect of changing or  influencing
          the control of the issuer of the  securities and were not acquired and
          are not held in connection with or as a participant in any transaction
          having that purpose or effect.

          Material to be Filed as Exhibits:

Exhibit A -    Copy of an Agreement  between Dr. Rosenwald,  Paramount  Capital,
               Aries Domestic and Aries Trust to file this Statement on Schedule
               13G on behalf of each of them.

Exhibit B -    List of executive officers and directors of Paramount Capital and
               information called for by Items 2-6 of this statement relating to
               said officers and directors.

Exhibit C -    List of executive  officers and  directors of Aries  Domestic and
               information called for by Items 2-6 of this statement relating to
               said officers and directors.

Exhibit D -    List of  executive  officers  and  directors  of Aries  Trust and
               information called for by Items 2-6 of this statement relating to
               said officers and directors.



                                        7





                                   SIGNATURES

                  After  reasonable  inquiry and to the best of my knowledge and
belief,  I certify  that the  information  set forth in this  statement is true,
complete and correct.


                                        PARAMOUNT CAPITAL ASSET MANAGEMENT, INC.

Dated:   July 23, 1998
         New York, NY                   By /s/ Lindsay A. Rosenwald, M.D.
                                           ------------------------------
                                               Lindsay A. Rosenwald, M.D.
                                               President


                                        ARIES DOMESTIC FUND, L.P.
                                        By Paramount Capital Asset 
                                           Management, Inc.
                                               General Partner

Dated:   July 23, 1998
         New York, NY                   By /s/ Lindsay A. Rosenwald, M.D.
                                           ------------------------------
                                               Lindsay A. Rosenwald, M.D.
                                               President


                                        THE ARIES TRUST
                                        By Paramount Capital Asset 
                                           Management, Inc. 
                                               Investment Manager

Dated:   July 23, 1998
         New York, NY                   By /s/ Lindsay A. Rosenwald, M.D.
                                           ------------------------------
                                               Lindsay A. Rosenwald, M.D.
                                               President


Dated:   July 23, 1998
         New York, NY                   By  /s/ Lindsay A. Rosenwald, M.D.
                                            ------------------------------
                                                Lindsay A. Rosenwald, M.D.


                                        8




                                    EXHIBIT A

                                    AGREEMENT

                          JOINT FILING OF SCHEDULE 13G

                  The undersigned hereby agrees to jointly prepare and file with
regulatory  authorities  a  Schedule  13G  and  any  future  amendments  thereto
reporting each of the undersigned's  ownership of securities of Enzon, Inc., and
hereby  affirm  that such  Schedule  13G is being filed on behalf of each of the
undersigned.

                                        PARAMOUNT CAPITAL ASSET MANAGEMENT, INC.

Dated:   July 23, 1998
         New York, NY                   By /s/ Lindsay A. Rosenwald, M.D.
                                           ------------------------------
                                               Lindsay A. Rosenwald, M.D.
                                               President


                                        ARIES DOMESTIC FUND, L.P.
                                        By Paramount Capital Asset 
                                           Management, Inc.
                                               General Partner

Dated:   July 23, 1998
         New York, NY                   By /s/ Lindsay A. Rosenwald, M.D.
                                           ------------------------------
                                               Lindsay A. Rosenwald, M.D.
                                               President


                                        THE ARIES TRUST
                                        By Paramount Capital Asset 
                                           Management, Inc. 
                                               Investment Manager

Dated:   July 23, 1998
         New York, NY                   By /s/ Lindsay A. Rosenwald, M.D.
                                           ------------------------------
                                               Lindsay A. Rosenwald, M.D.
                                               President


Dated:   July 23, 1998
         New York, NY                   By  /s/ Lindsay A. Rosenwald, M.D.
                                            ------------------------------
                                                Lindsay A. Rosenwald, M.D.


                                        9







                                                              EXHIBIT B

         The name and principal occupation or employment, which in each instance
is with Paramount Capital Asset Management,  Inc. ("Paramount  Capital") located
at 787 Seventh Avenue,  48th Floor, New York, New York, 10019, of each executive
officer and director of Paramount Capital is as follows:


                                          PRINCIPAL OCCUPATION
         NAME                             OR EMPLOYMENT
         ----                             -------------

Lindsay A. Rosenwald, M.D.       Chairman of the Board,  President of Paramount
                                 Capital  Asset  Management,   Inc.,  Paramount
                                 Capital Investments LLC and Paramount Capital,
                                 Inc.

Peter Morgan Kash                Director   of    Paramount    Capital    Asset
                                 Management,   Inc.,   Inc.,   Senior  Managing
                                 Director, Paramount Capital, Inc.

Dr. Yuichi Iwaki                 Director   of    Paramount    Capital    Asset
                                 Management, Inc., Inc., Professor,  University
                                 of Southern California School of Medicine

Item 2.

         During  the five  years  prior to the date  hereof,  none of the  above
persons  (to the best of  Paramount  Capital's  knowledge)  was  convicted  in a
criminal  proceeding  (excluding traffic violations or similar  misdemeanors) or
was a party  to a civil  proceeding  of a  judicial  or  administrative  body of
competent jurisdiction,  as a result of which such person was or is subject to a
judgment,  decree or final order enjoining future  violations of, or prohibiting
or mandating  activities subject to, Federal or State securities laws or finding
any violation with respect to such laws.

Items 3-6.

         Please refer to Items 3-6 herein reporting the beneficial ownership.


                                       10




                                    EXHIBIT C

         The name and principal  occupation or employment of the General Partner
of Aries Domestic, which is located at 787 Seventh Avenue, 48th Floor, New York,
New York, 10019, is as follows:

                                                      PRINCIPAL OCCUPATION
         NAME                                         OR EMPLOYMENT
         ----                                         -------------

Paramount Capital Asset Management, Inc.     General Partner; Investment Manager

         Exhibit B is hereby incorporated by reference.

Item 2.

         During the five years prior to the date  hereof,  the above  person (to
the best of Aries  Domestic's  knowledge)  has not been  convicted in a criminal
proceeding (excluding traffic violations or similar misdemeanors) or was a party
to a  civil  proceeding  of a  judicial  or  administrative  body  of  competent
jurisdiction,  as a result of which such person was or is subject to a judgment,
decree  or final  order  enjoining  future  violations  of,  or  prohibiting  or
mandating activities subject to, Federal or State securities laws or finding any
violation with respect to such laws.

Items 3-6.

         Please refer to Items 3-6 herein reporting the beneficial ownership.


                                       11




                                    EXHIBIT D

         The name and principal  occupation or employment,  which in the case of
Paramount Capital Asset Management,  Inc. is located at 787 Seventh Avenue, 48th
Floor,  New York, New York,  10019,  of each  executive  officer and director of
Aries Trust is as follows:

                                                            PRINCIPAL OCCUPATION
         NAME                                               OR EMPLOYMENT
         ----                                               -------------

Paramount Capital Asset Management, Inc.                    Investment Manager

MeesPierson (Cayman) Limited                                Trustee
P.O. Box 2003
British American Centre
Phase 3, Dr. Roy's Drive
George Town, Grand Cayman

         Exhibit B is hereby incorporated by reference.

Item 2.

         During the five years  prior to the date  hereof,  neither of the above
persons  (to the best of Aries  Trust's  knowledge)  have  been  convicted  in a
criminal  proceeding  (excluding traffic violations or similar  misdemeanors) or
was a party  to a civil  proceeding  of a  judicial  or  administrative  body of
competent jurisdiction,  as a result of which such person was or is subject to a
judgment,  decree or final order enjoining future  violations of, or prohibiting
or mandating  activities subject to, Federal or State securities laws or finding
any violation with respect to such laws.

Items 3-6.

         Please refer to Items 3-6 herein reporting the beneficial ownership.


                                       12