Yep, I would say alternator or maybe its drive belt is goosed. The charge light going out is conditional on an output from the alternator: the lamp earths through the winding until the alternator gives an output: ie when the engine starts the lamp goes out. So if lamp on when engine running, no output from alternator.
Check voltage at battery when engine running: should be above 14 Volts. A Nat Tyres or similar can test the battery and output for you.
I usually use the "Thank You" button