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


                                  SCHEDULE 13G

      INFORMATION STATEMENT TO BE INCLUDED IN STATEMENTS FILED PURSUANT TO
            RULES 13d-1(b)(c) AND (d) AND AMENDMENTS THERETO FILED
                           PURSUANT TO RULE 13d-2(b)

                               (AMENDMENT NO. 1)*


                           RENAISSANCERE HOLDINGS LTD.
- --------------------------------------------------------------------------------
                                (Name of Issuer)


                                  COMMON STOCK
- --------------------------------------------------------------------------------
                         (Title of Class of Securities)


                                    G7496G103
- --------------------------------------------------------------------------------
                                 (CUSIP Number)


- --------------------------------------------------------------------------------
             (Date of Event Which Requires Filing of This Statement)


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

               [X]    Rule 13d-1(b)

               [ ]    Rule 13d-(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 the
   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).

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- -------------------                                          -------------------
CUSIP NO. G7496G103                    13G                   PAGE 2  OF  5 PAGES
- -------------------                                          -------------------

- --------------------------------------------------------------------------------
   1      NAME OF REPORTING PERSON
          S.S. or I.R.S. IDENTIFICATION OF ABOVE PERSON

          OPPENHEIMER CAPITAL  (IRS NO. 13-3413767)
- --------------------------------------------------------------------------------
   2      CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP*            (a) [ ]
                                                                       (b) [X]

- --------------------------------------------------------------------------------
   3      SEC USE ONLY


- --------------------------------------------------------------------------------
   4      CITIZENSHIP OR PLACE OF ORGANIZATION

          DELAWARE
- --------------------------------------------------------------------------------
                             5     SOLE VOTING POWER
                   
                                   -0-
    NUMBER OF SHARES       -----------------------------------------------------
      BENEFICIALLY           6     SHARED VOTING POWER
        OWNED BY  
          EACH                     1,071,175
        REPORTING          -----------------------------------------------------
         PERSON              7     SOLE DISPOSITIVE POWER
          WITH 
                                   -0-
                           -----------------------------------------------------
                             8     SHARED DISPOSITIVE POWER

                                   1,071,175
- --------------------------------------------------------------------------------
   9      AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON


          1,071,175
- --------------------------------------------------------------------------------
   10     CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9)
          EXCLUDES CERTAIN SHARES*                                         [ ]

- --------------------------------------------------------------------------------
   11     PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)

          4.7
- --------------------------------------------------------------------------------
   12     TYPE OF REPORTING PERSON*

          IA
- --------------------------------------------------------------------------------

                      *SEE INSTRUCTION BEFORE FILLING OUT!


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ITEM 1         (a)    NAME OF ISSUER:
                      Renaissance Holdings Ltd.

               (b)    ADDRESS OF ISSUER'S PRINCIPAL EXECUTIVE OFFICES:
                      Renaissance House, 8-12 East Broadway
                      Pemmbroke HM 19 Bermuda

ITEM 2         (a)    NAME OF PERSON FILING:
                      OPPENHEIMER CAPITAL

               (b)    ADDRESS OF PRINCIPAL BUSINESS OFFICE:
                      Oppenheimer Tower, World Financial Center
                      New York, New York 10281

               (c)    CITIZENSHIP:
                      Not Applicable.

               (d)    TITLE OF CLASS OF SECURITIES:
                      Common Stock

               (e)    CUSIP NUMBER:
                      G7496G103

ITEM 3         IF THIS STATEMENT IS FILED PURSUANT TO RULE 13d-1(b), OR
               13d-2(b), CHECK WHETHER THE PERSON FILING IS A:

               (a)    [ ] Broker or dealer registered under Section 15 of the
                          Exchange Act;

               (b)    [ ] Bank as defined in Section 3(a)(6) of the Exchange 
                          Act;

               (c)    [ ] Insurance company as defined in Section 3(a)(19) of 
                          the Act;

               (d)    [ ] Investment company registered under Section 8 of
                          the Investment Company Act;

               (e)    [X] Investment adviser registered under Section 203 of
                          the Investment Advisors Act of 1940;

               (f)    [ ] Employee benefit plan or endowment fund in
                          accordance with Rule 13d-1(b)(1)(ii)(F);

               (g)    [ ] Parent holding company or control person, in
                          accordance with 13d-1(b)(ii)(G);

               (h)    [ ] Savings association as defined in Section 3(b) of
                          the Federal Deposit Insurance Act;

               (i)    [ ] Church plan that is excluded from the definition
                          of an investment company under Section 3(c)(14) of the
                          Investment Company Act;

               (j)    [ ] Group, in accordance with Rule13d-1(b)(1)(ii)(H).

               If this statement is filed pursuant to Rule 13d-1(c), check this 
               box.   [ ]


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ITEM 4         OWNERSHIP.

               (a)   Amount beneficially owned:  1,071,175 **

               (b)   Percent of Class:   4.7

               (c)   Number of shares as to which such person has:
                     
                     (i)   Sole power to vote or direct the vote:  -0-

                     (ii)  Shared power to vote:  1,071,175 **

                     (iii) Sole power to dispose or direct the disposition 
                           of:  -0-

                     (iv)  Shared power to dispose or direct the disposition 
                           of: 1,071,175 **

- ---------------
** This report is being filed on behalf of Oppenheimer Capital, a Delaware
   general partnership and/or certain investment advisory clients or
   discretionary accounts to report the fact that the reporting person has
   ceased to be the beneficial owner five percent or more of the securities of
   the issuer.

ITEM 5         OWNERSHIP OF FIVE PERCENT OR LESS OF A CLASS.
               
               If this statement is being filed to report the fact that as of
               the date hereof the reporting person has ceased to be the
               beneficial owner of more than five percent of the class of
               securities, check the following [X].

ITEM 6         OWNERSHIP OF MORE THAN FIVE PERCENT ON BEHALF OF ANOTHER PERSON.

               Not Applicable.

ITEM 7         IDENTIFICATION AND CLARIFICATION OF THE SUBSIDIARY WHICH ACQUIRED
               THE SECURITY BEING REPORTED ON BY THE PARENT HOLDING COMPANY.

               Not Applicable.

ITEM 8         IDENTIFICATION AND CLARIFICATION OF MEMBERS OF THE GROUP.

               Not Applicable.

ITEM 9         NOTICE OF DISSOLUTION OF GROUP.

               Not Applicable.


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ITEM 10        CERTIFICATION.

               By signing below I certify that, to the best of my knowledge and
belief, the securities referred to above were acquired in the ordinary course of
business and were not acquired for the purpose of and do not have 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.

                                    SIGNATURE

               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.


                                           Date: February 9, 1999


                                           /s/ THOMAS E. DUGGAN
                                           ----------------------------------
                                               General Counsel and Secretary




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