Warning: I've never troubleshooted a battery/alternator problem on an MB, so this is more just common troubleshooting tactics I'm telling you here.

Could be a bad battery, bad alternator, or bad connection somewhere. I'd start by checking all your connections for a good connection (remove, clean and re-attach) and check the belt tension (you'd probably hear a loose belt). Check your ground strap and check for continuity in all the common places to assure yourself the the battery voltage (engine not running) is the same at the battery as it is everywhere else you have a hot wire. If it's not, you've got a connection problem.

Once you do that, with the engine running check the voltage at the battery (you've got a good connection now right?). If it's not 12.5, then you probably have an alternator problem (or voltage regulator, not sure if that's integrated with the alternator or not). If it is 12.5, then you either fixed the problem with a bad connection or you still have the problem in which case it's a bad battery.

All that being said, I'm leaning towards your alternator or a bad connection since you initially noticed this problem with the engine running.

__________________
-justin

1987 300TD, 1987 300TD
2008 R32, 2000 Passat Wagon