GEN light on when engine is off

The other day the GEN light in my dash board began to light when I turned off the engine. So now when the engine is off the GEN light is on, but as soon as I start the engine this GEN light turns off.
I'm not sure what this should tell me? - is it an indicator for broken alternator or battery (both is pretty new)?, or is mess in some wires?

I hope someone can tell how to solve this...

Thanks!

My truck is a '65, 3.8 L

 

Update: today it will not start. The GEN light is still on, the battery seems charged but the engine is completly dead, when turning the key all I hear is at small click. Maybe it is an issue with the starter?

 

Check voltage regulator, it sounds as if the contacts are stuck in a closed position.

 

You may find something in these threads helpful. Most favor the regulator being the problem:

http://corvaircenter.com/phorum/read.php?1,532471,532565

http://www.impalas.net/forums/showthread.php?t=5679

http://www.chevelles.com/forums/showthread.php?t=332818

 

Thanks - I'll give the regulator a look...

 

Gen Light On

This indicates the regulator's cutout contacts are sticking or are welded closed .

The regulator is Kaputi and prolly the generator too , test it well before wasting
$ on a new regulator & frying it .