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 July, 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 June 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
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Of which total exercised by end of June 2004             0             0
-------------------------------------------------------------------------
Remainder                                        4'000'000    40'000'000
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                                   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:  July 5, 2004