I agree, and I suspect it might be draining back into the alternator due to a faulty rectifier.
Try lashing in an ammeter, but make sure you don't include the large cable that goes to the starter motor, just the others. Use heavy wire, and locate the ammeter where you can keep an eye on it.
You should see several amps on normal charge, any massive discharge from the battery will show up as "negative" current, i.e. the needle will go the wrong way.
If you don't see any discharge from the battery, but it still dies, I would replace the battery. It may have internal shorts. Does the battery get really hot when it discharges?