It could be that the battery cable that connects the battery to the logic board is faulty. Check for corrosion where the cable plugs in to the logic board. This happens a lot, in my experience. In this case, if the corrosion isn't too severe, a toothbrush on the affected area may help. If not, then you ''may'' need both the connector ''and'' a logic board. :-(
Or you could have gotten a defective battery.
Or your adapter is faulty.
If there is one in your area, Apple has a battery and adapter check that any reputable apple service authorized store/depot will do for free.