Interesting problem. I would try a couple of different things. First of all, even though I am pretty sure it is not your actual battery, I would try downloading "coconut battery" at:
http://www.coconut-flavour.com/coconutbattery/
It's free and this program will tell you what the original, current and maximum charge is of your battery. If that doesn't shed any new light on the situation. Then I would think your problem is either the battery connector or the dc in board. Both of which can be found with the following links:
http://www.ifixit.com/iBook-Parts/iBook-G4-14-Inch-Battery-Connector/IF184-001
http://www.ifixit.com/iBook-Parts/iBook-G4-14-Inch-DC-In-Board/IF184-003
Both parts are fairly inexpensive. If it isn't either of those, then I would assume the worst and say it is your logic board. Please let me know what you find out!