Age old question...Bad Battery or Bad Alternator???

Do this
Chances are if you have had a weak charge for a while the battery may be getting close to its death.I would bet it's the alternator if it's reading under 12 volts while running.

Yes, a good charging system will put out between 13.3 and 15 volts when the engine is running. If over 15 volts, your regulator is not doing its job.