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 May, 2005

                              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 reporting of newly created equity securities from conditional capital to
the SWX Swiss Exchange
In accordance with the reporting  requirements  pursuant to the Listing Rules 
(Art. 64 - 75) we have disclosed following information to the SWX Swiss 
Exchange:

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

The current capital structure of Converium Holding AG is as follows:

Spent equity capital                          Number of shares       CHF
Status as of the end of April 2005:           146'689'462            733'447'310
Status as at the end of the previous month:   146'689'462            733'447'310
Title category:
Conditional capital                           Number of shares       CHF
Number of exercised securities*               0                      0
Amount of remaining conditional capital*      4'000'000              20'000'000

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

This report is also made if no equity securities have been created from
conditional capital in the month in question.




                                   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/ Terry Clarke
                                                    ----------------    
                                                    Name:  Terry Clarke
                                                    Title: CEO


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


Date: May 23, 2005