If your "BATTERY" light is on while your engine is running, then
the problem is usually either the alternator or the voltage regulator.
This simple test will help you figure out which, without having to
remove anything. It applies to 60's vintage GM cars using a points-style
voltage regulator. Here is a picture of that type of regulator.
With the engine running, connect the negative lead of your voltmeter to
ground (or the negative terminal on the battery) and the positive lead to
the #2 terminal of the voltage regulator. If the #2 terminal is hard
to get a good reading from, you can instead measure the voltage
on the black wire that is in the 2-wire connector on the back
of the alternator.
If the voltage is under 5 volts, then the alternator is bad. (Or the belt tension is way too loose.)
If the voltage is over 5 volts, the problem is either the regulator or the
wire harness between the regulator and the starter solenoid. Most likely it
is the regulator. Since the
regulator is so easy to remove, you should just pull it and take it in to be tested.