2000 BMW 528 Won't start

I noticed yesterday morning that when I went to start the car, the battery sounded weak, but started. Over the course of the day, each time I started it, it sounded weaker and weaker finally to the point of not starting at all - just doing the clicking thing. My husband (an electrician) put a meter on the battery and said that it read 12 volts. He then jump started it with his truck and it started fine. We let it run for a little bit, then turned off and restarted which it did. That was last evening. When I went to start it this morning, it sounds real weak again.

Can it be the battery even if it still reads 12 volts or is it something else like the alternator? Thanks for you help.