Mitsubishi Repair: Possible issue with my Altenator? 96 Eclipse, battery cables, altenator

Question
I just purchased a 1996 Eclipse RS. 2.0 manual - with over 100 k miles on it.  The car will start just fine after sitting for a day or so.  However when I drive for about 15 min the battery light comes on and my radio turns off by itself.  Then when the car is shut off for even a couple hours and I try to restart it, it won't turn over but my alarm chirps uncontrollably...if i keep trying over and over after about 30 min it will finally start but still same issue with battery light.  I took it to Autozone and they tested the altenator and said it was fine but my battery was completely dead.  So I had them install a battery big enough to run an SUV.  The next day the exact same situation is happening again.  

Any idea what this could be? or how it can be fixed?
I would really appreciate any feed back.  Thank you for your time!

Answer
Jade,
It sounds like there is a problem with the charging system.  This can be caused by a short circuit, poor ground, or poor connection somewhere between battery and alternator, or the battery and starter.  Sometimes worn battery cables can prevent the proper amount of current from getting to the battery.  But I would also have the alternator double checked by a mechanic to be sure.  Most Mitsubishi alternators last between 7-10 years and tend to show a lot of signs before they die completely.  Though AutoZone felt your alternator was fine, it still may be the overall problem.  Have it and the starter checked by your mechanic.
Good luck!