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It goes to the heater core housing on the firewall inside the car. It's right above the passenger's feet.
You'll have to take the passenger side kick panel off (several 7mm screws). Then you have to take the ECM out (a couple of 10mm bolts).
Agent 86 did a nice post on how to do a heater core replacement with lots of pictures. He has the C68 electronic HVAC system, so it's slightly different than what you have. See this post:
The 2nd and 3rd pictures show the motor that the C68 systems use. Your cable ends at that point and there is a "L" shaped lever there that does the same function (moves the blend door to control the heat).
The 4th picture shows what it looks like with the motor removed. You'll have a bracket there with the L shaped lever. The bracket holds the cable housing and the cable connects to the L shaped lever.
You can't zoom the bracket and lever but you can the cable. This is the bracket and lever attached to the heater core case:
http://www.ebay.com/itm/1984-1989-Corvette-A-C-Control-Cable-Bracket-/370534190464?pt=Motors_Car_Truck_Parts_Accessories&hash=item56458f0d80&vxp=mtr
This is the cable, the red tab is attached to the AC control and the other end of course to the lever and bracket assy:
http://www.ebay.com/itm/1984-1989-Corvette-Heater-Control-Cable-GM-/371026290541?pt=Motors_Car_Truck_Parts_Accessories&hash=item5662e3e76d&vxp=mtr
Here's the case where the bracket/lever assy. attaches to, the rod extending out of the case attaches to the lever:
I would doubt it's the cable, the lever or the control and more than likely one of two clips at the heater box. The clip/retainer that holds the rod to the temp door is a GM 20007898 (14) and that can easily be found and there are numerous substitutes.
The other that attaches the rod to the lever in the eBAy link is more difficult to find GM 1224062 (4). That's a difficult find but again there are substitutes.
Here's a case the bracket, lever, rod and both clips (4,5,7,13,14)