Acer One Aspire ZG5 battery connection issue?

I have a problem with my Acer One Aspire Zg5. Sometimes it behaves as if there is no battery attached, so it will run with the cord attached, but as soon as the cord is unplugged, it dies instantly. No lights, nothing. Then sometimes the battery suddenly works, and it charges and discharges for a few days - it will work normally. Then suddenly it stops working and the Acer behaves as if there is no battery attached at all. While the cord is disconnected, there's no light, and nothing happens when I press the power.

It's flipped back and forth between the two states a few times. It's like there's a bad connection somewhere. But I don't know how to test it. Could be the battery, or maybe the terminals in the Acer.

Any suggestions?

Thanks,