CERTIFICATION PURSUANT TO RULE 30A-2(B) UNDER THE 1940 ACT AND SECTION 906 OF
                             THE SARBANES-OXLEY ACT


I, James A. Bowen, Chairman of the Board,  President and Chief Executive Officer
of  First  Trust  Enhanced   Equity  Income  Fund   (Formerly   known  as  First
Trust/Fiduciary Asset Management Covered Call Fund) (the "Registrant"),  certify
that:

         1.     The Form N-CSR of the Registrant  (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 Registrant.

Date:    FEBRUARY 26, 2008                /S/ JAMES A. BOWEN
     ----------------------               -----------------------------------
                                          James A. Bowen, Chairman of the Board,
                                          President and Chief Executive Officer
                                          (principal executive officer)


I, Mark R. Bradley,  Treasurer,  Controller,  Chief Financial  Officer and Chief
Accounting Officer of First Trust Enhanced Equity Income Fund (Formerly known as
First  Trust/Fiduciary  Asset Management Covered Call Fund) (the  "Registrant"),
certify that:

         1.     The Form N-CSR of the Registrant  (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 Registrant.

Date:    FEBRUARY 26, 2008               /S/ MARK R. BRADLEY
     ----------------------              ---------------------------------------
                                         Mark R. Bradley, Treasurer, Controller,
                                         Chief Financial Officer and Chief
                                         Accounting Officer
                                         (principal financial officer)