2004 Ford Mustang Will Not Hold a charge

My car will not start. I can jump it, and it will take a charge. I can drive it, but once I turn it off, it will not re-start.
When running, the voltmeter indicates the system is charging.
While it is running, electrical devices like the radio will "cut in and out". Sometime just the rear speakers cut out, then come back.
The battery still has a "green" dot to indicate it is OK.