EXHIBIT 32.1

                            CERTIFICATION PURSUANT TO
                  SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

In connection with the Annual Report of St. Jude Medical, Inc. (the "Company")
on Form 10-K for the period ended December 31, 2003 as filed with the Securities
and Exchange Commission on the date hereof (the "Report"), I, Terry L. Shepherd,
Chief Executive Officer of the Company, certify, pursuant to 18 U.S.C. ss.1350,
as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that:

     1.   The Report fully complies with the requirements of Section 13(a) or
          15(d) of the Securities Exchange Act of 1934; and

     2.   The information contained in the Report fairly presents, in all
          material respects, the financial condition and results of operations
          of the Company.


                                        /s/ TERRY L. SHEPHERD
                                        ------------------------------------
                                        Terry L. Shepherd
                                        Chairman and Chief Executive Officer
                                        March 12, 2004