That is the problem, as I say the light needs to be on when ignition is switched on, it's glowing when running as the alt changes state from being an earth to giving a positive charge hence this is what extinguishes the light when charging, the bulb is working so must have a connection but I would still make sure the connection is good alt end first but I suspect you have a problem with the ignition feed to the bulb. Do all other dash lights etc work ok?

If you rev the car to around 4000 RPM this will start the charge process as well.

Martin