Right on the money. The alternator is not charging the battery.While the car is running, the battery should read 14 volts+, it sounds like an alternator problem.
Perform load test on battery and alternator. This problem will not go away without a new battery or alternator(possibly voltage regulator).
If you have the time disconnect the battery cables and put the battery on a good charger for a few hours at 10-20 amps.
If the battery comes up to 12+ volts it may not need a battery, but it is really not the best way to test it. A charging system load tester is the best way to go.