I am having the same problem with a 63 equipped with the original 413. It starts, runs for a short time, and dies. The problem is that it seems to be electrical in my case. The carb is getting gas, I have switched to an electric fuel pump, replaced the coil, ballast resister, condenser, and points. I have however noticed that the top part of the breaker plate that secures the points and condenser is loose, and I am attempting a remedy for that. The fuel flows, but the points can shift, and it seems that this has gotten worse till now when the car is totally unreliable.
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