I agree with Heyinway, sounds like a poorly rebuilt alternator, or bad voltage regulator (internal). With a meter on the battery, your should be getting 14-16v when you idle up. Also, check all major grounding points, battery to body, engine block to body, etc., for looseness or corrosion.
Battery voltage is too low, it may have a bad cell.
I've had new batteries that tested bad before installation. Engine off...fully charged battery voltage should be close to 13 volts with a new battery...no matter where in electrical system you check. Engine on...turning on all 12 volt items, the alternator should constantly maintain no less than what the battery is with engine off.
Ensure that alternator has a good ground. Alternator is putting out minimum voltage...and likely not much amperage. This failure can happen when alternator is trying to charge a dead battery...alternator internal components overheat and fail. In your case, it probably a bad rebuild to start with.
Anything 12 volt that is trying to operate at less than 12 volts and enough amperage is going to overheat and fail. Same happens with too high voltage and amperage. It has to be within a narrow range of both.
replaced engine to body ground
Problem solved