Do you have the older points distributor? If so, check the points gap; it should be around .017-.018" gap. Too small, and it will put excess current into the coil and ballast.
Now for the big questions: What PN of ballast are you using? And what is the coil model?
Does the car start and then blow the ballast? Then check as above at the battery as the car starts: see if the voltage is going way above 14 volts after it starts.
Also, check the voltage at the blue wire to the ballast with the car OFF. Make sure that there is 0 volts when OFF. If not, then what may be happening is that the ballast is staying hot and blowing while the car is off (and the points are closed all the time). When you go to start the car, the ballast bypass function of the ignition switch in the START position puts power to the coil and the engine will start, but then not stay running when you release the key from START to RUN since the ballast is already blown.
BTW, normal voltages at the ballast when running are .3 to .5 volts or so below the battery voltage on the blue wire, and 6-10 volts from ground at the wire from the ballast to the coil +.