Yes, it is the starter solenoid (as opposed to the master solenoid) welded closed, it must be replaced. This is a hazardous situation but the engine will not run with the keys out of the key switch, unless there are other failures too. None the less, a good reason to perform routine shutdown mag checks (at idle power please), and never turn on the master when someone/something is in the prop arc. If the aircraft has been found with a welded starter contactor, the starter motor should be examined, and maybe replaced. It is possible that it continued running for the entire flight following the previous start. There may be pinion and ring gear wear and damage, overheating damage, and brush wear. It would be difficult to determine that such inspection is not necessary, and it's not burdensome, so it should just be done. This is a good reason for a starter engaged light, which is required in some locales.
In the case where you see the starter engaged light prior to takeoff, shut down and investigate. For a Lycoming, you could safely taxi back to the apron to shut down. For a Continental O-300 or larger, the engine should be shut down immediately, and the aircraft towed back.