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                                    Form 6-K

                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                        Report of Foreign Private Issuer

                        Pursuant to Rule 13a-16 or 15d-16
                     of the Securities Exchange Act of 1934

                           For the month of June, 2004

                              CONVERIUM HOLDING AG
                   ------------------------------------------
                 (Translation of registrant's name into English)

                                 Baarerstrasse 8
                                   CH-6300 Zug
                                   Switzerland
             -------------------------------------------------------
                    (Address of principal executive offices)

Indicate by check mark whether the registrant files or will file annual reports
under cover Form 20-F or Form 40-F.

                              Form 20-F X       Form 40-F
                                     -------             -------

Indicate by check mark whether the registrant by furnishing the information
contained in this Form is also thereby furnishing the information to the
Commission pursuant to Rule 12g3-2(b) under the Securities Exchange Act of 1934.

                                    Yes          No X
                                       -------    -------

If "Yes" is marked,  indicate the file number assigned to the registrant in
connection with Rule 12g3-2(b): 82- Not Applicable







Monthly disclosure of equity securities newly created from conditional capital

Issuer:                  Converium Holding AG
Sec. ID no.:             1'299'771
ISIN:                    CH0012997711

In accordance with art. 3a of the Articles of Incorporation of Converium Holding
Ltd and with para. 5.02 (conditional capital) of the checklist published as
enclosure 1 of the "Reports required in order to maintain a listing" (Art. 64 -
75 KR), we inform about the creation of following share capital:

Newly created share capital
As at end of Mai 2004                                No new shares
As at end of previous month                          No shares

Contingent capital                                   No. of shares  CHF
Reserved for Option Rights and/or Conversion Rights  4'000'000      40'000'000
Of which total exercised by end of Mai 2004          0              0
Remainder                                            4'000'000      40'000'000




                                   SIGNATURES

Pursuant to the requirements of the Securities Exchange Act of 1934, the
registrant has duly caused this report to be signed on its behalf by the
undersigned, thereunto duly authorized.


                               CONVERIUM HOLDING AG




                               By:  /s/ Dirk Lohmann
                                    Name:      Dirk Lohmann
                                    Title:     CEO




                               By:  /s/ Christian Felderer
                                    Name:      Christian Felderer
                                    Title:     General Legal Counsel



Date:  June 3, 2004