Why is the battery not charging?

I have a 2.0 ghz Core 2 Duo (Merom) White Macbook. It refuses to charge the battery now. When plugged in, the Magsafe adapter does not light up at all, but the battery icon changes to Not Charging. I have tried three different adapters with the same results. I have also tried two different known-good batteries (1 Apple OEM and 1 aftermarket) without any improvement.

I can still use the computer with only the power cord. If I remove the battery, the computer continues to function.

If I remove the power cord but leave the battery in, the computer will continue to function. The battery refuses to charge.

Any suggestions? It was charging perfectly overnight and has just stopped charging.