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Assuming it's the same as my '77, you need to unhook the cable behind the dash. There's a small spring type lock holding it in. You push that in towards the dash while pulling out on the cable. I can get mine out without removing or loosening the dash, but others can't always get to it. You might need to loosen the up the dash a bit. From there you just slide it right out.
I'd take a peek up under the dash and try to see how the cable routes under there. For me, whenever I need to do any work under the dash, the first thing that comes out is the hard plastic duct from the center out to the lower left vent, then you can see a lot more under there. You'll have to pop out the grommet where the speedo goes through the firewall. It's probably old a brittle, I'd just figure on buying a new one, they aren't that expensive. I take channel locks to unscrew it from the tranny. If it's a 4 speed, it will be on the passenger side.
If your car is automatic with cruise control, then you have two cables. The short one comes from the speedo to the transducer (located on the wheel well adjacent to the brake booster) and the longer one from the transducer to the transmission.
I posted the same question on the forum about a month ago. After reading many of the suggested methods, I found that the best way was to remove the A/C duct work from under the dash on my 1975. You need to have someone pull the cable through the firewall while you push down on the cable release tab behind the speedo. The fun part is lining-up the new cable at the trans connection. Good luck.
If your car is automatic with cruise control, then you have two cables. The short one comes from the speedo to the transducer (located on the wheel well adjacent to the brake booster) and the longer one from the transducer to the transmission.
If you have speed control you have two cables. From my experience the transducer to trans cable is more likely to break. Simply unscrew the bottom cable from your transducer (located under your brake booster) and look at the end going into the transducer. If it is intact and with flat sides, it is probably OK. Take the top cable off and examine the end as before. If it looks OK, spin it with your fingers and have someone look if the needle moves. If either cable is broken, you don't need to replace the outer sheath, just the inner cable. Your GM dealer will charge you $32 for the same universal cable you can buy at Autozone for $7. This will replace the broken inner cable, if one is broken on yours. If you need to go this route, put some JB Weld on the end before crimping it. This is good insurance it won't come loose. If you need to remove the cable from the speedo, just take your time and locate the back. You can use a pencil eraser tip to push in the retaining clip on the back while someone pulls from the outside. The pencil gives you some reach in a tight space.
My 75 has one cable but its two cables hooked together the Factory splice is right before the transmission and be carful it does push up in side the tranny take your time slipping it back together I was using the grease cable deally and it actualy worked.