Starter circuit trouble

[WilKo@aol.com]

Hey there,
My BN6 had this happen: I hit the starter button and the starter moved and 
stayed on but the car didn't start, and with the ignition key turned "off", the 
starter just wound down and took the weak battery with it. During the time it 
was dying I tried jiggling the button and even pushed it a few times to no 
avail. 
Does this sound like a case of a starter solenoid stuck in the "on" 
postition?

Rick-not good with electrical
San Diego