I have a buddy whos got battery issues with his Integra. The battery drains out after a very short period of time (less than a day). We boost it, it's fine. When it's running, it's fine. Starting the car after 5 or 6 hours can be a problem. It drains out eventually to a point where it doesn't start.
Can't be alternator because it charges it when running, neither can it be the charging harness for that same reason.
My guess is the battery itself pretty much crapped out. Can anyone confirm?