The alternator should provide sufficient voltage to run without the battery, so I doubt the battery is the source of your trouble. However, if it's old, draining it this low can push it over the edge, so you may end up having to replace it anyway.
Jump start it and attach a DMM to the battery. If it reads anything less than 12.6 volts, you're going to need an alternator.
A cool little trick you can try if you don't have a DMM is to use a screwdriver. If the alternator is working properly, a slight magnetic field will be created at the rear of the alternator when it's running. If the screwdriver sticks to the back of the alternator, then it's working.
My money says it's bad.
Take care
~Chris
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