DC9 CREW; ON FINAL; LOST RELIABLE GS INDICATION IN #1 ADI.

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FLT #XXX MSP-BIL WAS FLOWN ROUTINELY TO BIL ON 01/01/THUR. WX HAD BEEN RPTED AND FORECAST AS CLR AND 10 SM VISIBILITY. NO ALTERNATE FUEL WAS BOARDED; BUT 46 MINS OF CONTINGENCY FUEL WAS ON BOARD. AT BIL; THE APCH CTLR RPTED THAT VISIBILITY WAS DETERIORATING. WE HAD BRIEFED FOR A VISUAL APCH TO RWY 28 WITH THE POSSIBILITY OF AN ILS. WE WERE AT 11000 FT WHEN APCH SAID VISIBILITY WAS NOW 2 MI AND WE WERE GIVEN A DSCNT CLRNC AND THEN A L TURN VECTOR TO INTERCEPT AND CLRNC FOR STRAIGHT-IN ILS RWY 28. THE RADIOS WERE TUNED AND IDENTED FOR THE ILS AND THE ADI GS INDICATORS SHOWED WE WERE ABOVE THE GS. GEAR DOWN AND 40 DEGS FLAPS WERE SELECTED AND A RAPID DSCNT ON THE LOC WAS COMMENCED. APCHING 5000 FT; THE FO STATED THAT HE SHOWED WE WERE DSNDING BELOW THE GS. MY GS INDICATOR ON MY ADI WAS STILL SHOWING FULL DEFLECTION DOWN; BUT MY FLT DIRECTOR V-BARS WERE RISING. I STOPPED THE DSCNT AS WE GOT AN AURAL 'GS' WARNING AND THE TWR CALLED TO SAY THEY HAD AN ALT WARNING ON US. I INITIATED A MISSED APCH AND WE WERE VECTORED FOR AN APCH TO RWY 10L. I THEN REMEMBERED SEEING A PREVIOUS WRITE-UP IN THE LOGBOOK FOR THIS PROB IN WHICH THE PLT RAPPED ON THE ADI TO UNSTICK THE GS INDICATOR; AND MAINT HAD CLRED BY TESTING AND SAYING THEY COULD NOT DUPLICATE. WE SHOT THE APCH TO RWY 10L AND LANDED AS NEW WX RPTED AT 1/2 MI VISIBILITY AND 100 FT BROKEN (LCL GND FOG). FLT INSTS ALL INDICATED NORMALLY. A RADIO REPAIR STATION REPAIRMAN CAME OUT TO THE AIRPLANE AND TESTED THE ADI AND VERIFIED THAT IT HAD STUCK IN THE DOWN POS AND THE CONDITION WAS INTERMITTENT. THIS INCIDENT WAS AGGRAVATED BY ENCOUNTERING UNFORECAST BAD WX WITH A SUDDENLY LOW FUEL CONDITION; NO CURRENT WX RPTS FOR NEARBY ARPTS; A LAST MIN APCH CLRNC WHILE HIGH AND SEEING A LCL FOG CONDITION DEVELOPING OVER THE ARPT AND FEELING URGENCY TO GET ON THE GND BEFORE THE WX WENT BELOW MINIMUMS.