Battery light not going out.

[krzracingcouk]

i have posted this on Facebook too to see if anyone can shed some light:-

I have a '82 Coupe GT 5s, when I start the engine the battery light stays on normally for a while (not sure if right for this age of car) on that day but every start after that goes out fairly quickly (about a minute or so) and the voltage on the volt meter from about 11v to 13v ish. The other day the battery light did not go out and the battery went flat on a drive.
I have changed the alternator for a different one and put a meter on the battery and voltage does go up when engine is running but battery light stays on and VDO gauge does not agree with meter directly on battery. Is there some relay that is playing up somewhere? Is there some checks I can do to see what is not working? Have also wondered if it is worth running a ground lead directly from the body of the alternator to the -ve of the battery?
I look forward to any help.