EXHIBIT 32

    CERTIFICATION BY THE CHIEF EXECUTIVE OFFICER AND CHIEF FINANCIAL OFFICER
                       PURSUANT TO 18 U.S.C. SECTION 1350,
      AS ADOPTED PURSUANT TO SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

- --------------------------------------------------------------------------------

I, Richard B. Handler, Chief Executive Officer, and I, Joseph A. Schenk, Chief
Financial Officer, of Jefferies Group, Inc, a Delaware corporation (the
"Company"), each hereby certifies, 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 Company's periodic report on Form 10-Q for the period ended June 25,
2004 (the "Form 10-Q") 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 Form 10-Q fairly presents, in all
material respects, the financial condition and results of operations of the
Company.

                                      * * *

CHIEF EXECUTIVE OFFICER                           CHIEF FINANCIAL OFFICER

  /s/ Richard B. Handler                            /s/ Joseph A. Schenk
- -------------------------------                   ------------------------------
      Richard B. Handler                                Joseph A. Schenk

Date: July 30, 2004                               Date: July 30, 2004

A signed original of this written statement required by Section 906 has been
provided to Jefferies Group, Inc. and will be retained by Jefferies Group, Inc.
and furnished to the Securities and Exchange Commission or its staff upon
request.

                                     Page 1