I was driving the other day and all of a sudden the speedometer needle started jumping form 70 to 80 to 60, etc and then finally settled on zero and stayed there. I'm guessing that it is a broken cable. The car is a 1968 327/300 with AT. How big a project is it to replace the original cable? Any tips would be appreciated.

 

There are a few points of failure for those symptoms; but it doesn't necessarily mean "bad news". Usually, when the needle jumps, then speedo quits, the problem is a broken speedo cable. If there are no other problems with the cable 'jacket', replacing the [inner] cable should resolve the problem with no damage to the speedo head.

Order the replacement cable first, unless you can leave your car in a "waiting" mode. You also need a tube of dry graphite powder [lube].

Reach under the driver's dash with right arm and find the rear of the speedometer. Where the cable attaches, there is a little squeeze-clamp; squeeze it to release the cable, and while squeezing, pull the cable from the speedometer. Pull the end of cable down to gain some access, then pull the entire cable out of the jacket. You may want to put some rags/towel on the floor to catch any graphite lube from the old cable.

Verify that the new cable is the same length and has same ends as the old one. Put a 'shot' of dry graphite lube on a rag/paper towel in your hand. Now lay one end of the new cable in the lube and draw it slowly thru the lube to put graphite powder on the entire cable. You don't need a lot on it...just make sure it has a 'wipe' of it all the way down.

Insert the transmission end into the sheath and push the cable all the way down. Push & turn the cable a bit to get it to find position in the speedo drive gear inside the transmission. The end ferrule should then be seated in the 'bell' end of the sheath. Now, squeeze that retainer clip to open it and wiggle the cable into the speedo. Release the clip and all is done.

IF the cable actually broke in two pieces inside the sheath, you will be able to remove the outer portion. To remove the other piece, you will have to remove the cable from the transmission (unscrew the end piece to remove it], then use the other half of that cable to push the remaining piece out of the sheath. (Cut the frayed end off so that it will fit into the sheath.)

 

The 68 speedometer has a screw on cable rather than the squeeze clamp as in the 1969 models.
Just reach under and behind the speedometer and unscrew the cable and do the same at the transmission. you can pull the cable out through the firewall from the engine side and then pull it up from in front of the transmission one you unscrew it from the transmission. Quite simple.
You can also just pull the inter cable out from the sheath once you take it loose from the trans and speedometer. Just leave the cable in place and slide the old inter cable out and put the new one in.
Just follow your prior members instructions on lub of the cable.
The main thing to know is the 68 has the screw on and off cable. No squeezing anything except maybe you wife, girlfriend, or mistress.