EXHIBIT (32b)

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”) Annual Report on Form 10-K/A for the period ended June 27, 2004 as filed with the Securities and Exchange Commission on the date hereof (the “Report”), I, William M. Lowe, Jr., Vice President, Chief Operating Officer and Chief Financial 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:

(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: March 31, 2005	By:	/S/ WILLIAM M. LOWE, JR.
		William M. Lowe, Jr.
		Vice President, Chief Operating Officer and      Chief Financial Officer