Electrical Trouble?

[bongfoo]

ok here the deal, i have a 92 corolla LE. i just bought it like a week ago. today my wife was using the car with no problems. then around 6 this evening the car wouldn't start. so we got a jump from our neighbor and it started to run for. she started to go down the street and the lights went dim and the car died. naturally i assume it is the alternator and so i remove it and have it tested. it was only putting out about 12.9 volts. so i purchased and installed the new alternator. i get a jump from my friend and let the Battery charge. but this is where it gets freaky..... as i get ready to drive it down the street i turn on the light and the whole car just died on me! so we do alittle testing. i left the lights on and hooked up the jumpercales to my friends car and the lights got brite. they stayed that way for about 3 minutes and they dimmed. all of the connections are clean and tight i just can't figure out what the heck is going on! any suggestions? i think i may take the Battery out and have that tested but beyond that i really don't know what it could be!