St. Jude Medical (STJ) 10-Q 2004 Q1 Quarterly report

Exhibit 32.1

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

In connection with the Quarterly Report of St. Jude Medical, Inc. (the “Company”) on Form 10-Q for the period ended March 31, 2004 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. §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

May 7, 2004