[bret a ewing]

Silk,
If the electrical aspect has been eliminated:
When the car "dies out" if it happens to be at low rpm or idling the problem could be fuel flooding caused by high fuel pressure. This would effect the idle circuit of the carburetor. Pushing fuel past the needle and seats. After the car stalls, even more fuel pours into the carb, which may explain the hard to start issue. If you do not have a fuel regulator (that works) on your system this type of problem may occur.