From what you said it sounds like there is a problem with wiring on the back side of the alternator. On most cars the battery warning light supplies power to the exciter field of the alternator and that lead should have close to 12 volts on it under normal operation. If the lead voltage goes low for some reason the light will turn on and the alternator will not be able to generate any power to charge the battery.
You might try removing the rear connector to the alternator and then start the engine to see if the battery warning light still lights up. If it does then there is a problem on the field lead. It may have a short to ground on it.