Gerry,
Do you have power to the coil when the key is in the "run" position? I would suspect a problem between the coil and the fuse box.

 

As would I. In particular, the wiring at the ballast resistor on the coil. The resistors have been known to burn out as well.

 

But if it runs when the starter is engaged, that suggests that somehow the power is getting from the black wire (the starter solenoid wire) to the ignition. Having had the ignition switch apart not long ago, I can see how that might happen. Try pulling the connection below the ingition switch (three wires coming from the switch to the connector: red, green and brown and a separate black wire from the switch that goes to the starter solenoid) and see what happens when you turn the key to the run position and the start position. In the run position the red and brown wires should show no resistance, and in the start position the black and red should also show no resistance. Another possibility is that since the starter solenoid is getting power the CS solenoid is also activated, and the engine is running on the fuel from the CS solenoid. (Is that possible?)

 

Sounds like a bad ignition switch... But I am probably wrong...

 

But if it runs when the starter is engaged, that suggests that somehow the power is getting from the black wire (the starter solenoid wire) to the ignition.

There are two separate power sources for the coil. One source is from ignition switch terminal #16 (green wire). This provides the coil with battery voltage during cranking (which is why the engine runs when the starter is engaged). This is known as the ballast resistor bypass circuit.
The other power source starts at the ignition switch terminal #50 (brown wire), goes to the top of fuse #4 (or is it #5?) where it connects to a green wire with a black stripe. This green/black wire then goes to one side of the ballast resistor. The other side of the resistor connects to the coil positive. This is the run circuit.
My thoughts; Since you say there is fuel, that means that the ignition switch can be ruled out because the fuelpump is powered thru the brown wire from the ignition switch. That leaves, as sprintgtc said, a problem between the fusebox and the coil. I'd double the the wiring at the coil/ballast resistor. There's a wiring diagram in the electrical section of this BB. If the wiring is okay, use a test light or voltmeter to test the run circuit (green/black wire) to determine where the open is.

 

And there's the difference between experience and seat of the pants. My Spider has electronic ignition, and so the green wire from the ignition switch doesn't go anywhere - three wires go into the connector, but only two come out. Let us know what the final result is.