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                        AGREEMENT APPOINTING A SUCCESSOR             Exhibit 4.1
                      RIGHTS AGENT UNDER RIGHTS AGREEMENT


           AGREEMENT, dated as of June 1, 1999, among Service Corporation
International, a Texas corporation (the "Company"), Harris Trust and Savings
Bank ("Harris Trust") and The Bank of New York ("BNY").

           The Company and Harris Trust have heretofore executed and entered
into the Rights Agreement, dated as of May 14, 1998 (the "Rights Agreement"),
under which Harris Trust is the Rights Agent. Pursuant to Section 21 of the
Rights Agreement, the Company may appoint a successor to the Rights Agent in
accordance with the provisions of Section 21 thereof. All acts and things
necessary to make this Agreement a valid agreement, enforceable according to
its terms, have been done and performed, and the execution and delivery of this
Agreement by the undersigned have been in all respects duly authorized by the
undersigned.

           In consideration of the foregoing and the mutual agreements set
forth herein, the parties hereto agree as follows:

           1.   Effective as of June 1, 1999, Harris Trust resigns as Rights
                Agent under the Rights Agreement and the Company appoints BNY
                as the Rights Agent under the Rights Agreement.

           2.   Harris Trust and the Company each hereby waive any notice
                required by Section 21 of the Rights Agreement.

           3.   BNY represents that it, together with its Texas subsidiaries
                and affiliates, meets the requirements to be Rights Agent as
                set forth in Section 21 of the Rights Agreement. BNY accepts
                its appointment as Rights Agent under the Rights Agreement and
                agrees to act as Rights Agent in accordance with the terms of
                the Rights Agreement.

           IN WITNESS WHEREOF, the parties hereto have caused this Agreement to
be duly executed and attested, all as of the date and year first above written.




Attest:                                   SERVICE CORPORATION INTERNATIONAL

                                       
By: /s/ Curtis G. Briggs                  By:    /s/ James M. Shelger
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                                               Senior Vice President


Attest:                                   HARRIS TRUST AND SAVINGS BANK

By: /s/ Lorraine Rodewald                 By:  /s/ Ray G. Rosenbaum
    ------------------------------------       ----------------------------------------------
                                               Vice President


Attest:                                   THE BANK OF NEW YORK

By: /s/ Steven R. Myers                   By:  /s/ James Dimino
    ------------------------------------       ----------------------------------------------