INATTENTION TO SWITCH POSITION WHILE BALANCING FUEL WING TO WING RESULTS IN FLAME OUT OF RIGHT ENGINE FOR B737-500 FLT CREW.

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WHILE CRUISING AT FL370; I HAD A FUEL IMBAL BTWN THE MAIN TANKS. ORIGINALLY THE #2 ENG WAS HEAVIER THAN THE #1 BY 300 LBS. I GOT CAUGHT UP TELLING A STORY AND THE NEXT THING I KNEW WAS THAT THE #1 MAIN WAS NOW 500 LBS HEAVIER THAN THE #2 MAIN. I DIDN'T PAY ATTN TO MY VALVES IN CHANGING THE XFEED. I MUST HAVE TURNED OFF THE XFEED VALVE WHEN I TURNED OFF THE #2 MAIN TANK PUMPS. WITHIN ABOUT 10 SECONDS; THE #2 ENG FLAMED OUT. WE BROKE OUT THE QRH; ADVISED ATC OF OUR NEED TO GET AN EMER DSCNT; AND ATTEMPTED TO RESTART THE ENG MULTIPLE TIMES. WE FINALLY HAD TO START THE APU AND USE APU-ASSISTED PROCS TO GET THE ENG STARTED WHEN PASSING FL260. I CONTINUED TO DSND TO FL240; OUR ASSIGNED ALT. I SENT DISPATCH A MESSAGE ADVISING THEM OF WHAT HAPPENED. I CHKED MY POS -- I WAS 175 MI E OF ZZZ1 AND 225 MI S OF ZZZ. IN MY OPINION; ZZZ (MY ORIGINAL DEST) STILL QUALIFIED AS THE NEAREST SUITABLE ARPT BECAUSE TO ME THE DIFFERENCE IN DISTANCE WAS NOT THAT SIGNIFICANT. I SENT DISPATCH A MESSAGE SAYING THAT IT WAS MY INTENT TO CONTINUE TO ZZZ UNLESS TOLD OTHERWISE. THEY CONCURRED. LNDG AT ZZZ WAS UNEVENTFUL. PAY ATTN TO SWITCH POS! MY INATTN LED TO A SIGNIFICANT EVENT THAT SHOULD NOT HAVE HAPPENED. I SHOULD HAVE TAKEN MORE TIME TO ANALYZE THE PROB AND TAKEN INTO CONSIDERATION THAT A MALFUNCTION CONTRIBUTED TO THE FLAMEOUT. IF I HAD DONE THAT; I WOULD HAVE CHOSEN ZZZ1 AS NEAREST SUITABLE FOR THE FACT THAT IT WAS CLOSER IN STRAIGHT LINE DISTANCE AND THAT IT HAS MAINT.