I have a 1995 Monte Carlo z34. I have installed a new battery and alternator but the battery will still die out while the engine is running. I placed a meter on the battery and watched it initially draw a charge when it reached 10.5 volts. It would charge up to 12.3 volts and then begin discharging again. This was normal and went on for about 20 minutes of run time and then it would simply die off to 9.3 volts and the engine would stall. I let the car sit for an hour, started it, and went through this all over again. Does anyone have any ideas?

Sounds like you are not charging at all. A properly functionaing alternator should yield you something in the neighboorhood of 14.5volts. In you case I would check wiring connectings on alternator, and possibly take a voltage reading right on the connection there. If you don't see about 14.5 here and your belt is not slipping I would suspect a defective laternator.
Note: not all "new" alternators are good; and a completely dead battery can kill a good alternator.
Good Luck,
Morrowrj