The voltage across the battery was 3.1 micro-V, current draw across the circuit was 2 micro-A when system was unplugged and 0 when power supply was plugged into the wall. I'm beginning to suspect it's the PS, because the modem (brand new) has been having problems connecting to AOL (the modem it replaced would not connect at all, but another computer at the same location connects to AOL first time, everytime). I'll have to run some tests on the PS. If anyone has any other ideas, I'm listening.
BTW, the antivirus is current, so I'm quite sure it's not a virus.