3M Company Form 10-Q dated March 31, 2004, Exhibit 32.1

EXHIBIT 32.1

SARBANES-OXLEY SECTION 906 CERTIFICATION

In connection with the Quarterly Report of 3M Company (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, W. James McNerney, Jr., Chief Executive Officer of the Company, certify, pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that, to my knowledge:

(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/ W. James McNerney, Jr.

W. James McNerney, Jr.

Chief Executive Officer

May 4, 2004