Exhibit (32)a

      Certification of Chief Executive Officer
Pursuant to Section 1350 of Chapter 63 of Title 18 of the United States Code

I, Thomas J. Falk, Chief Executive Officer of Kimberly-Clark Corporation, certify that, to my knowledge:

(1) the Form 10-Q, filed with the Securities and Exchange Commission on May 7, 2004 (“accompanied 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 accompanied report fairly presents, in all material respects, the financial condition and results of operations of Kimberly-Clark Corporation.

/s/ Thomas J. Falk
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Thomas J. Falk
Chief Executive Officer
May 7, 2004

Exhibit (32)b

      Certification of Chief Financial Officer
Pursuant to Section 1350 of Chapter 63 of Title 18 of the United States Code

I, Mark A. Buthman, Chief Financial Officer of Kimberly-Clark Corporation, certify that, to my knowledge:

(1) the Form 10-Q, filed with the Securities and Exchange Commission on May 7, 2004 (“accompanied 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 accompanied report fairly presents, in all material respects, the financial condition and results of operations of Kimberly-Clark Corporation.

/s/ Mark A. Buthman
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Mark A. Buthman
Chief Financial Officer
May 7, 2004