You might want to try an SMC reset. I would say it was a magsafe board issue but you stated that the power adapter light is green, if it were a magsafe board issue the light would not be on. Then again I could be wrong, But I have never heard of a magsafe board being faulty and still emitting light. My next guess would be that the battery connecter within the computer has malfunctioned. Is this an Aluminum MacBook?
Does the computer work from the power adapter only and without the battery? From what I see, the part is relatively cheap, however, I don't want to steer you in the wrong direction. If you live near an Apple store, I would recommend getting it looked at, as they will diagnose it free, despite it being out of warranty (Assuming it is). Here is a link to a connector that you could compare to yours. I'm sure if you do some research, there will be someone else out there with a similar issue if not the exact same issue. I will keep doing research and let you know if I come up with anything,
With 860 cycles on a removable battery you're well past the 300 cycles that Apple states for average battery life. As Upland13 says take it into an Apple store if you have one close and have it checked. If no Apple store is close do you have a friend with a white or black MacBook? You could try their battery in yours to see if it's the battery or some other problem.
There’s one problem with running your MacBook with the battery removed. MacBooks and MacBook Pros automatically reduce the processor speed by about half when the battery is removed.
If you have a MacBook with the removable battery you can get a new third party battery fairly cheap on eBay.
Or for $129 from Apple http://store.apple.com/us/product/MA566LL/A?mco=MTY3ODQ5OTY