Bad solenoid?

My son's friend has a 1998 Jetta with the V6 and manual transmission. Suddenly today it will not start. The starter will not spin or click or anything. You can push start it so the ignition system is fine. We found good voltage to the starter through the high amp wire. We then pulled the wire going to the solenoid and when turning the key to the start position we get the full 12+ volts. This would indicate to me that the solenoid is bad. But a complete starter is less expensive so we would replace the whole thing. Sound like we are on the right track?