To me it sounds like either the alternator or the charging control circuit is faulty. If the alternator belt was slipping however, you would be able to hear it, apart from the fact that you would also be able to smell the rubber melting. One way to confirm is to put a voltmeter across the battery terminals. It should read about 12 volts. When the engine is running, it should go up to around 13 or 13.5 volts which basically indicates the battery is being charged.