The Li-Ion batteries from a 3400 or original G3 (often called the Kanga or 3500) will fit in a 5300 and even power it, but they cannot be recharged by the 5300 and probably won't be recognized correctly if at all.

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Are you sure it not a 3400 or G3 battery?

I understand that less than 100 PowerBook 5300s shipped with LiIon batteries, and they were all recalled. Of course, recalls never attain 100% compliance, so a few batteries may have slipped through. You would have, for all practical purposes a one of a kind item if you really have a 5300 LiIon battery.

More info on the batteries: http://support.apple.com/kb/TA21834?viewlocale=en_US