Now that I read your your post again, I might have misunderstood it:

When the car stops, are you still trying to crank it with the starter?
Or has it just started, you've released the key and now the car stalls?
When you say that the dash lights go out, you mean they go out as they
normally would when the car is running?

If the car is cranking normally, I was wrong suggesting the starter or
related issues so forget what I said in that case.
John's suggestion is very valid. A convenient way to check the
running/start position is to put a multimeter on any fuse that is
powered while the car is running. The window wiper or cruise control,
both under the dash, are good examples of fuses to use for this
purpose.
Note I am not suggesting that these fuses are blown: just that they
should have 12V when the car is running/starting. If your car stops and
the voltage disappears on these fuses, it means that the ignition
switch or connecting wires might be your problem.
If the voltage is still present and the car stops running, perhaps
troubleshoot the ignition or fuel supply (possible main relay).

Just making sure -- wouldn't want to steer you in a wrong direction.
Regards,
Remco