EXHIBIT (32a)

CERTIFICATION PURSUANT TO 18 U.S.C. SECTION 1350,
AS ADOPTED PURSUANT TO
SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

In connection with the Unifi, Inc. (the “Company”) Quarterly Report on Form 10-Q for the period ended March 28, 2004 as filed with the Securities and Exchange Commission on the date hereof (the “Report”), I, Brian R. Parke, Chairman and 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, as amended; and

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

Date: May 12, 2004	By:	/s/ BRIAN R. PARKE
		Brian R. Parke
		Chairman and Chief Executive Officer