What you should do is check the info reported by System Profiler for your battery. Check under the 'Power' heading on the left hand side and look at the battery stats - particularly the number of cycles, full battery charge and the health. I'm guessing the health will be poor and the full charge will not be too high - don't have mine handy for a reference I'm afraid. If your cycle count is low enough, you may be able to persuade an Applecare rep to replace it for you free of charge. 300 is usually the magic number. If your battery is below this, you have a case.
Feel free to cite these numbers to Applecare, though I would prefer you avoid showing exactly where you got them. 'The internet' should be a good enough answer if they ask.
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