DISPATCHED WITH THE LEFT GENERATOR DEFERRED INOP; B737-400 MUST DIVERT WHEN APU GENERATOR GOES OFF LINE DUE TO AN OVERSPEED.

Narrative

ACFT ARRIVED AT THE GATE WITH MAINT ALREADY ON THE ACFT. MAINT RPTED THAT WHILE TAXIING THE ACFT TO THE GATE FROM A REMOTE POS; THEY WERE UNABLE TO GET THE #1 ENG DRIVEN GENERATOR ON LINE. AFTER ASSISTING THE MECHS WITH AN ENG RUN; THE #1 ENG GENERATOR WAS CONFIRMED TO BE INOP. MAINT CTL DECIDED TO PLACE THAT GENERATOR ON MEL. THE FO AND I REVIEWED THE PROCS FOR THAT MEL AND DISCUSSED THAT WE NEEDED TO OPERATE THE APU DURING THE ENTIRE FLT TO HAVE THAT GENERATOR PWR THE COMPONENTS FROM THE INOP #1 ENG GENERATOR. ENG START AND CLBOUT TO CRUISE ALT WERE ROUTINE. DURING CRUISE; WHILE MONITORING THE OUTPUT OF THE APU GENERATOR; I NOTICED THAT THE FREQ OF THAT GENERATOR WAS VERY HIGH. I SENT AN ACARS MESSAGE TO MAINT INFORMING THEM OF THAT INDICATION. APPROX 10 MINS LATER; THE APU GENERATOR FELL OFF LINE WITH AN OVERSPD INDICATION. NOW; DOWN TO 1 REMAINING ACFT GENERATOR; WE DECLARED AN EMER WITH ATC AND DECIDED TO DIVERT TO ZZZ; WHICH WAS ABOUT 80 MI AHEAD OF US. WE COORD WITH THE FLT ATTENDANTS AS WELL AS THE COMPANY TO ADVISE THEM WHAT WE HAD DECIDED. WX WAS NOT A FACTOR AND ATC GUIDED US DIRECTLY TO THE ARPT WHERE WE LANDED WITH NO FURTHER INCIDENT. CFR WAS ON THE GND TO SUPPORT US IF NEEDED. WE COMPLETED THE TAXI TO THE GATE WHERE WE TURNED THE ACFT OVER TO MAINT AFTER HAVING MADE A WRITE-UP IN THE LOGBOOK.