I agree with everything Dana wrote in the previous post. Since you wrote ballast resistor singular, I'll assume this is a points igniton for the moment.
The dark blue wire from the voltage regulator to the alternator is fried and a blue wire from the resistor ballast looks like she was smoking too but not melted.
Blue wire is the "ignition run" and labelled J2 on the wiring diagrams.
Ignition run is switched on by the key and supplies power to the coil, and the alternator field. A branch goes to the voltage regulator so it can sense what voltage the system is at.
Melting wire indicates a short. Most likely shorted to ground. Follow the melted blue wire right up the the end of where ever it goes. That's going to be close to the short. If the wire has chafed against some metal - that will be the cause. It's melted at the terminal, remove the terminal and brush. Maybe it lost its insulator.
The ignition run circuit look something like this. Whether you car has a welded splice as shown, or the wires are joined at a terminal, you'll have to tell us.