Catherine is correct
The source of the problem sounds like one of three things.
1) DC powertap is damaged
2) DC charging circuitry on motherboard is damaged - in some cases, this is a daughterboard which is replaceable without replacing the motherboard, but most of the time it is a function of the motherboard itself and requires replacing the entire unit.
3) The battery itself is bad and is causing the DC charging circuit to prematurely turn off AC input.
If you plug the AC adapter in, and watch the leds on the front of the unit (pay attention to the one which indicates if the system is on AC power) wiggle the plug slightly and see if the led turns off, then back on. IF this happens, it is your DC powertap. Don't wiggle so much that you break a perfectly good tap.
If you find the DC powertap is not at fault, the next thing to do is remove the battery and see if the unit runs reliably on AC power only. If it does run fine on AC power only, then it is either the charging circuity or the battery. Try using a "known-good" battery and see if the symptoms continue.
If with a "known-good" battery the symptoms continue it is in fact the charging circuitry and warrants evaluation of replacement of motherboard (possibly a dc-dc daughterboard) or replacement of the laptop outright.
If you find that the problem is the DC powertap, the parts are cheap, but the labor is costly. Typically $10-20US for the parts, and 2-2.5 hours labor to repair.
If you feel comfortable performing the repair yourself, then great, but if not I suggest you find a local family-owned service shop to perform the work. If you can't find one, email me back and I may be able to find someone to help you.
If you do take the unit in for repair, please remove the harddrive as it is not necessary to perform this work (personal privacy issue). If you need access to your data while the system is being worked on, you can easily install the harddrive into a usb enclosure ($30US) so that you can access your data on another system.
Best of success