Have you tried starting it up at night with the lights all off?  This will show sparking if there is any that you haven't seen.

Some of the guys who are really good at this will be dropping in soon.

Good Luck!

My suggestion would be to replace the coil. The "cloth covered" wire is the resistor wire for the coil. It is designed to provide reduced voltage to the coil while the engine is running, thereby keeping the points from welding themselves together. If that wire is getting hot, and the smoke is leaking out of it, that is an indication that there is too much resistance in the circuit. The only thing in that circuit is the coil.
Here's a check for you to do before sending it out to Mr. Goobwrench... If you have a voltmeter, check the voltage on the "+" terminal of the coil while cranking the engine. You should see a full 12v. If you only see like, 9, or even 6 volts, you have lost the 12v power for "start" from the starter. There is a wire that comes from the "S" terminal of the starter, that gives the coil 12v only while cranking. If this wire is damaged/disconnected, the coil won't get the full 12v it needs to start the car. This would also overpower the resistor wire, and might cause it to overheat.
So, if you determine that there is 12v at the + terminal while cranking, I would replace the coil.

I had it running monday and pulled each wire at the distributor cap to see if sparking was occurring. All wires checked out OK.

WHen I check for voltage to the coil, should the wire from the distributor to the negtive side of the coil be disconnected? Last night, I connected the positive lead to the positive side and the negative to the ground or block, the wire to the distributor was disconnected and I also jumped a ground from the negative to the block. When I turned the ignition switch to the on position (not cranking), I was only getting 0.678 volts. I know that something is wrong, here....HELP!

You're running a wire from the "-" side of the coil to a ground?!?!? That don't sound right to me.
The "-" coil terminal should only have the wire going to the distributer, and possibly to the capacitor on the side of the coil.
You shouldn't have a full 12V at the "+" terminal unless you are cranking it. But... are you sure it was .678 volts, or 6.78 volts?
Key, on, engine off voltage on the + terminal should be in the 6-8 volt range. It shouldn't matter if the - side of the coil is connected.

Upon reading the post a little more carefully, I see that maybe you meant you were connecting the ground of the meter to the block... My bad!
Still, please clarify the reading as .678, or 6.78...big difference there. If it is 6.78, then that would be right, if not, then Houston, we have a problem!
My best bet would be the resistor wire itself.

If I confirm the 0.678 volts,  I was thinking about changing out the ignition switch. What do you think?

It may well be the switch, but a more likely source would be a connection somewhere. Have you attempted to run a 12v jumper to the + terminal of the coil to see if it will fire off and run? This would tell you(and us!) for sure the problem is in the power to the coil. Even if you do only have the .678 with the key on, you should still have the full 12V while cranking that comes from the starter. Have you checked the voltage while cranking? If the voltage is not correct while cranking, as well as key on/engine off, then I might be more inclined to go with the ign switch.
I gots tuh go back tuh werk now....but I'll keep me eyeballs on this one.

If you are talking about changing out the ignition switch that the key goes into, if it is on the column, then it won't help.  The switch on the column drives a rod to the switch box down by the bottom of the steering column on the upper left side of the column itself.

If the switch is onl the dash, then there are wires running up to the back of the switch.  I don't know where it is located on a 70, on a 71 it is on the column.

'68 was the only Shark to have the key in the dash. Those had the ign switch as part of the key cylinder...all one piece.
The '70 will have the actual ign switch fastened to the lower part of the column.