Intermittent stalling - won't restart (video inside)

[750hp]

I've been driving the Cobra every day for the last year and it hasn't missed a beat. This past week, I didn't drive it at all. On it's first start after the little break, the battery sounded a bit weak. I got 3 starts out of it and then the battery seemed dead. I clutch started it and got it home to the battery charger and it was fine throughout the following day. Today, the car runs for anywhere from 3 seconds to a couple of minutes but then cuts out. It's not a gradual feeling as though the carb is running low on fuel before finally stopping.

This stops instantly like someone had killed the ignition.

It takes some cranking to try to get it to refire after it stalls. The difference being, if I turn off the key to stop it, it will restart instantly.

Here's a video of it starting perfectly, but then only running for a short while before being unable to restart again.

[ame]https://www.youtube.com/watch?v=B3e0aYk6Em0[/ame]

I changed the battery and that didn't fix it.
I have fuel pressure at the carb.
I have fuel in the carb. (It's an Edelbrock carb)
It's not flooding at idle.
When it's running, it's running perfectly.

I heard that ignition switches can fail. I've tried running the car today with the headlights on to see if the dash lights do anything funny when it stalls. Everything seems normal.

Engine vacuum doesn't fluctuate in the split second it takes to stall.
Voltage at the gauge is around 12V and then 14V when the alternator kicks in. It can still stall when the gauge is at 14V, though it seems to happen less when there's RPM, engine load and higher voltage. I'm not sure which of those seem to help the situation.

It's a basic old tech car with a carb and an old Accel distributor. Even if both batteries had failed, I'd have thought that the engine would still run OK at 12-14V and not stall.

I'd love to hear your ideas!