I had a similar problem. I had a bit of crud lodged where the float needle seats in the carb. It would allow the bowl in the carb to slowly fill up when the car was off, but once you started it, the engine drained the bowl faster than it could fill up again, so the car would only run on the gas that was in the bowl and then die. Once the car died, the bowl slowly filled up again while I was scratching my head, and the car would start again for a short period of time.
My bet is a piece of crud in your needle. Remove the carb from the car, remove the one bolt on bottom that holds the thing together and inspect your needle and seat.