It sounds like the DC-In Board is the culprit. You can find a replacement DC-In Board [http://www.ifixit.com/iBook-Parts/iBook-G4-14-Inch-DC-In-Board/IF184-003|here] and the replacement guide [http://www.ifixit.com/Guide/Repair/Installing-iBook-G4-14-Inch-1-42-GHz-DC-In-Board/726/1|here].
I don't think it is a bad battery, because if only the battery were bad, then the laptop should still boot while plugged in. It is possible that the problem could lie with the logic board, but the DC-In is cheaper and (in my experience) more likely the problem.
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Have you tried using a different AC Adapter to power the laptop with?
It sounds like the DC-In Board is the culprit. You can find a replacement DC-In Board [http://www.ifixit.com/iBook-Parts/iBook-G4-14-Inch-DC-In-Board/IF184-003|here] and the replacement guide [http://www.ifixit.com/Guide/Repair/Installing-iBook-G4-14-Inch-1-42-GHz-DC-In-Board/726/1|here].
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I don't think it is a bad battery, because if only the battery were bad, then the laptop should still boot while plugged in. It is possible that the problem could lie with the logic board, but the DC-In is cheaper and (in my experience) more likely the problem.
It sounds like the DC-In Board is the culprit. You can find a replacement DC-In Board [http://www.ifixit.com/iBook-Parts/iBook-G4-14-Inch-DC-In-Board/IF184-003|here] and the replacement guide [http://www.ifixit.com/Guide/Repair/Installing-iBook-G4-14-Inch-1-42-GHz-DC-In-Board/726/1|here].