From your description, you have more than one problem. The warning light gets power from the battery, which is grounded at the alternator (or voltage regulator in your car) when the car is not running. When the car is started, the voltage produced by the alternator cancels out the battery voltage and the light goes out. If the alternator is putting out less voltage than the battery, the light will come on dimly. If the alternator fails, the light will come on and stay on.
If you are reading more voltage from the alternator than the battery holds, the warning light should not illuminate. Unless there is a lot of resistance in the line from the voltage regulator to the light, in which case the voltage at the light would be less than the battery voltage, causing it to light up.
I had this happen when an old solder joint in the wire from my alternator to the warning light corroded with age. The corrosion increased the resistance in the line, causing the light to illuminate and causing a faulty low voltage reading to the alternator. Reading this low voltage, the alternator put out more voltage to charge the system; in my case it went to more than 19 volts.
Check the resistance on the line running from the voltage regulator to the "ignition light." Resistance in this wire will trick your alternator into putting out too much voltage. This condition fried my voltage regulator and a battery.