A LIGHTNING STRIKE FOLLOWED BY AN ENG FLAMEOUT DURING APCH TO AZO; MI.

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AT APPROX 7500-8000 FT; WE WERE DSNDING THROUGH A CUMULUS CLOUD LAYER. WE HAD BEEN CLRED TO INTERCEPT THE ILS RWY 35 LOC AND DSND TO 2800 FT MSL. ONCE IN THE CLOUDS WE GOT AN 'ICE' MESSAGE AND PUT WINGS AND COWLS ON. SHORTLY THEREAFTER; THERE WAS A BRIGHT FLASH OF LIGHT ALONG WITH A LOUD BANG THAT SEEMED TO HIT THE ACFT DIRECTLY IN FRONT OF ME. WHILE SCANNING THE EFIS/EICAS FOR ANOMALIES; I IMMEDIATELY SAW (AND FELT) THE #1 ENG FAIL. N1 AND N2 WERE ALMOST BACK TO ZERO. HOWEVER; INTERSTAGE TURBINE TEMP REMAINED HIGH. WE GOT A MASTER WARNING DUE TO LOW OIL PRESSURE AND THE ACFT BEGAN TO YAW TO THE L. I IMMEDIATELY PUT IN THE APPROPRIATE CONNECTIONS AND NOTIFIED AZO APCH THAT WE WERE DECLARING AN EMER AND THAT WE WERE SINGLE ENG. I CALLED FOR THE APPROPRIATE EMER MEMORY ITEMS FOLLOWED BY THE CHKLISTS (QRH). I WAS THE PF AND REQUESTED FROM ATC TO LEVEL OFF AT 5000 FT AND GET VECTORS FOR A BOX PATTERN SO WE COULD COMPLETE OUR CHKLISTS AND SET UP FOR A SINGLE ENG APCH. I ELECTED NOT TO TRY A RESTART AS I SUSPECTED THAT THE ENG MAY BE SEVERELY DAMAGED. DUE TO OUR LOW ALT WE WERE UNABLE TO CONTACT OUR DISPATCH THROUGH ACR RADIO. NO ONE ON THE GND AT AZO OPS ANSWERED US EITHER. I TOLD AZO APCH TO PLEASE CONTACT OUR DISPATCH AND NOTIFY THEM OF OUR EMER STATUS. THE FO RAN THROUGH ALL THE NUMBERS AND WE DETERMINED THAT WE HAD PLENTY OF RWY AT AZO USING RWY 35. WE DID R TURNS BACK AROUND FOR THE ILS TO RWY 35. DURING THIS TIME I KEPT GETTING A 'RWD AUTOPLT' CAUTION MESSAGE; STATING THAT THE AUTOPLT WAS NOT KEEPING UP WITH THE TRIM REQUIREMENTS. I KEPT HAVING TO MANUALLY RE-TRIM. THIS ADDED GREATLY TO MY WORKLOAD. FINALLY; ON SHORT FINAL; I TURNED OFF THE AUTOPLT AND HAND FLEW THE ACFT TO TOUCHDOWN. AZO ARPT RESCUE AND FIRE FIGHTING FOLLOWED US AND CONFIRMED THERE WAS NO SMOKE OR FIRE OBSERVED. I DECIDED NOT TO EVAC BUT TO MAKE A SLOW TAXI TO THE GATE. WE SHUT DOWN AT THE GATE WITH NO FURTHER INCIDENT. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: CONTRACT MAINT FOUND NO PROBS WITH THE ENG. IT WAS DIAGNOSED AS GAS PATH FAILURE BROUGHT ON BY THE SHOCK WAVE OF THE LIGHTNING STRIKE. SEVERAL HOLES WERE BURNED INTO THE SKIN OF THE ACFT. CONTINUOUS IGNITION WAS NOT ON AND WAS NOT REQUIRED TO BE ON IN THE STATED WX AND ICING CONDITIONS.