Got a 2000 Cougar that has a charging problem, new alternator new battery, but kill the battery again, check the fuse on the cable thats connected to alternator its good, so charge the battery by jumping it and run idle a for while and all the guages shut down the engine down but when you have the booster cable it stays running with battery light, check all the fuses look good.

THE SAME THING HAPPENED TO MINE. AFTER $400 I REALIZED IT WAS THE FUSE WHICH IS CONNECTED TO THE ALTERNATOR. IT WAS BROKEN AND WOULD COME LOOSE AT ANY SLIGHT BUMP, THEN IMMEDIATELY KILL THE BATTERY. FIRST THE RADIO WENT OUT THEN THE DASH LIGHTS. ANYWAY I GOT A NEW ONE FOR $40 AND PROBLEM SOLVED. =]

Had the same problem. it ended up being the pigtail wire, the red wire end had opened up so it wasnt touching the contact

Two things you have to double check:

a) Is this the correct battery for this vehicle
b) Is this the correct alternator for this vehicle

If you answered yes then move onto the next item:
c) Are battery cables in good condition.
d) Are the wires from the rear of the alternator in good condition. (no corrosion , or looseness)
e) while the car is running with the charger off put a voltmeter( digital) on the battery and see how many volts it puts out with and without accessories on.
Gotta be between 13.5 volts to 14.5 volts. If no its the alternator. Remember as long as this is the correct battery and alternator for this vehicle. Its gotta be one of the items I've told you about. My experience with this vehicle is the wiring off the alternator.