#1 ENGINE FLAMEOUT RESULTED IN DIVERSION TO MCI. EMERGENCY DECLARED.

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AT FL310 THE #1 ENG WOULDN'T ADVANCE ABOVE 94% N1. F/O WAS FLYING. HE WAS ASSIGNED ALL FLYING; ATC DUTIES AND S/O AND I WORKED ON ENG PROB. WE CHKED FOR ANY APPLICABLE CHKLISTS (NONE). ALL OTHER ENG PARAMETERS WERE NORMAL. WE TRIED ENG A/I; IGN ON; #1 GEN AND BLEED AIR OFF; SWITCHED FUEL SOURCE; WITH NO CHANGE. WE MATCHED THE #2 TO 94% N1 AND ALL PARAMETERS MATCHED. SHORTLY AFTER ENG #1 DROPPED BACK TO 87% N1. AT THIS TIME S/O WAS CONTACTING COMPANY MAINTAINENCE. WE DESCRIBED SITUATION TO THEM AND THEY SAID STANDBY. ALSO CONTACTED COMPANY TO FIND OUT ABOUT DIVERTING TO MCI (KANSAS CITY). SHORTLY AFTER THIS; #1 ENG SLOWLY (30-40 SECS) SPOOLED DOWN TO IDLE POWER. AT THIS TIME WE CONTACTED CENTER AND DIVERTED TO MCI. RAN ENG FAIL CHKLIST. ENG QUIT SHORTLY AFTER THIS. ATC WAS VERY HELPFUL; AS MUCH FOR QUICKLY GETTING US A TURN TOWARDS MCI AND A LOWER ALT (FL240) AS FOR NOT ASKING US A LOT OF QUESTIONS OR BOTHERING US AT A BUSY TIME. DSCNT AND APCH TO MCI UNEVENTFUL. CREW PERFORMED WELL. F/A HANDLED THEMSELVES PROFESSIONALLY; BRIEFING WAS MADE TO FLT SERVICE MGR WHO THEN BRIEFED REST OF CREW. NO SUGGESTIONS; SYS WORKED AS IT SHOULD HAVE. CALLBACK REVEALED THE FOLLOWING INFO: WELD IN THE MAIN FUEL LINE TO THE FUEL CTLR SEPARATED CAUSING THE ENG TO FLAMEOUT. THIS HAS NOT BEEN A MAJOR PROB WITH THE ACFT. EMER WAS DECLARED AND RECEIVED FULL COOPERATION FROM THE CTLR FOR THE DIVERSION TO MCI AND LOWER ALT FOR ACFT PERFORMANCE. CAB; #1; WAS ADVISED OF THE DIVERSION AND THE ENG HAD BEEN SHUTDOWN BUT IT WAS NOT NECESSARY TO PREPARE THE CABIN FOR EVAC.