SMA PLT HAS DME FAIL ON VOR DME APCH.

Narrative

RWY 16 VOR/DME APCH TO PROVIDENCE; RI. WX -- IMC; 600 FT RPTED CEILING (RAGGED) 3 MI VISIBILITY; LIGHT RAIN AND FOG. AS I DSNDED THROUGH 700 FT; I NOTICED THE DME DISPLAY TO GO BLANK. WHILE ATTENDING TO THIS; TWR ASKED ME MY HDG; I NOTICED 220 DEGS; TWR INSTRUCTED IMMEDIATE MISSED APCH; I NOTICED DSCNT BELOW DECISION HEIGHT AS I EXECUTED MISSED APCH. THE WX HAD BEEN FORECAST AS 1100 FT CEILING AND 3 MI VISIBILITY. I ELECTED TO DO THE APCH EVEN THOUGH IT WAS BELOW MY PERSONAL MIN AS I DID NOT THINK TO ASK FOR THE ILS RWY 5; WHICH I WOULD HAVE BEEN COMFORTABLE WITH. THIS WAS MY FIRST FLT IN MODERATE-HVY IMC WITHOUT ANOTHER PLT SHARING THE WORKLOAD (I USUALLY FLY WITH MY WIFE WHO IS A PLT). WHEN THE DME FAILED; I FIXATED ON THAT PROBLEM AND FAILED TASK #1 'FLY THE PLANE.' THE NEXT FLT I DISCOVERED THAT THE ATTITUDE INDICATOR WAS UNRELIABLE; VACUUM WAS OK! THIS MAY HAVE LED TO THE APPROX 60 DEG R TURN. IN RETROSPECT; I SHOULD HAVE EXECUTED AN IMMEDIATE MISSED APCH WHEN THE DME FAILED; NOT TRIED TO FIX IT! TWR HAD ME TO THE ILS 5; NO PROBLEM.