There are relays in under the cover behind the ammeter. When you hold the "start" switch the engine will turn over. There is a relay that responds to oil pressure that keeps power to the ignition as long as there is pressure.
I seem to recall that there is also a delay relay that ignores no oil pressure for the first 30 seconds while the pressure builds. That might be on other models, it may depend on you holding the "start" switch.
Start it with the cover off of the relay box, and see if one of the relays releases just before the engine starts to die.
It may be as simple as a plugged or bad oil pressure switch.
Now that I think about this, if you hold the "start" switch continuously, does the starter crank forever, or does the oil pressure cause the relay to release the starter motor once it is running a bit?
Someone said "Making good decisions comes from experience, experience comes from bad decisions." I say there are three kinds of people: those who learn from their mistakes, those who learn from the mistakes of others, and those who never learn.