Engine starts, but dies within seconds

I have a 1985 Ford E150 Conversion Van with approximately 65K miles on the 5.0L engine. The engine starts up immediately every time, but dies just as quickly. Sometimes it will run for as much as 30 seconds before it quits, but that is rare. I rebuilt the E5TA YA 2BBL carb in an attempt to fix the issue, but the problem didn't change at all. I took the top off of the carb to watch fuel delivery and everything looks fine there too. Plugs, wires, cap, etc., are all new. Typically I would consider this a fuel pump, but the carb bowl is full when the engine dies and pressure from the pump seems quite high.