How To O-Ring Your C-3 Calipers
Please Note: This site is not endorsed by Vette Brakes Products. I do not work for Vette Brakes, and I am not compensated by Vette Brakes. But I do recommend their products, as my experiences with their company has been first class.
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C-3 rear calipers are notorious for "pumping air" into the hydraulic system. This can occur if the rotor run-out is out of spec, or if there is a bearing issue. Vette Brakes offers a kit that will allow you to rebuild your calipers with O-rings.

If you don't feel comfortable tackling this job, Vette Brakes also sells complete calipers and complete car sets.

Before starting, make sure your calipers are stainless sleeved. The photos below will help you to identify this. Also, if your caliper bores are badly scored, they may not be rebuildable.
Remove the caliper.  I'm using a breaker bar.
Lift and carefully support the vehicle. Then remove the wheel. Here, I'm working on the rear, as the rear brakes are notorious for pumping air.

Two bolts secure the brake caliper. I had to use a breaker bar to remove mine.
Remove the brake line.
Gently and carefully remove the brake line. Make sure the wrench fits securely over the nut. This is also a good time to inspect your lines. If they compromised in any way, replace them.

Make sure you don't get brake fluid on any painted part, as it will ruin any painted piece it touches.
Remove the bleeder.
As with the brake fluid line, gently remove the bleeder, making sure the wrench makes secure contact with the bleeder. Inspect the bleeder for any sort of damage. They cheap and easy to replace, if needed.
Remove the caliper bolts and separate the caliper.
Remove the two caliper bolts and separate the halves. Inspect the bolts for damage, rounding off, or stripping. Replace as necessary.
Remove the pads by pulling the cotter pin and rod.
Remove the cotter pin and the brake pad pin. Again, inspect the hardware and replace as necessary. Also, make note of the pad orientation so you'll be able to easily reassemble.
With the pistons still in the bores, clean the outside.
With the pistons still in the bores, clean the exterior of the caliper. Here, I'm using my parts washer. I use mineral spirits and a stiff wire brush to scrub all the dirt, rust, and grime off the caliper. Be gentle around the piston assemblies themselves.

But make sure to clean this area just the same.
Disassemble the pistons.
Disassemble the pistons. Each caliper has four.

To remove the piston, first gently push the piston into its bore. Then, with a screwdriver, gently pry up on the metal surround that holds the rubber seal that surrounds the piston.

Behind each piston, is a spring. When rebuilding, you can include this spring, or leave out. The purpose of the spring is to ensure gentle brake pad pressure to the rotor, at all times. I left mine out, as the spring was badly corroded and I did not want this constant contact.

The choice is yours. However, if your spring is in bad shape, replace it.

You'll notice my calipers are not pristine. My car is more of a driver, than a concours trailer queen, or a show car. It's your option to detail and repaint the calipers. If you go this route, make sure the paint is made for brake calipers, and you don't allow any paint to touch the mating portions of the calipers.
Here is the piston bore.
See how the piston bore is nice and shiny where the rest of the caliper is not? The nice silver circle tells me these calipers are sleeved with stainless inserts. At this stage in most Corvette's lives (a 1977 with over 100K) it's safe to assume most have received stainless sleeved calipers. But still, check to be sure.

If the bore is corroded or scored, it cannot be used. Gently run your finger around the bore. You should not feel anything.

Take a clean, soft cloth, soak it in clean brake fluid, and clean the bores. After you've cleaned the bores, rinse them with clean brake fluid.
Insert the new caliper half o-ring.
Your Vette Brakes kit will come with a small rubber O-ring. This replaces the caliper half O-ring. If you need help keeping it in place, you can place a small amount of bearing grease on it to make it stick. I did not, as my O-ring stayed in place.
Assemble the large O-rings to the new pistons.
On the left is a newly O-ringed piston. On the right, is the piston. Dunk the piston in clean brake fluid. I use an old (clean)jar filled with (clean) brake fluid. Also dunk the O-ring in the brake fluid. Then assemble.
Dunk the assembly in clean brake fluid.
Lots of product placement, right? I don't work for Ragu or Vette Brakes. And I'm not compensated by them. I do, uh, work for FedEx, however...

Dunk the new piston assembly in your clean jar of clean brake fluid. Then do the same with the rubber piston seal.

Gently take the rubber piston seal (with the gold round metal gasket) and wrap it around the piston. If you have questions about this, look at the orientation of your old piston assemblies. The large portion of the piston goes into the bore first, with the dimpled portion pointing up.

Make sure everything is well lubricated with clean brake fluid.
Gently tap the assembly into the bore.
Using a large socket, and a small, lightweight hammer, gently tap the metal gasket seal into the bore. Run your finger around the circumference. The metal gasket should be evenly seated around the bore, with no portion of it higher than another.

Remember, tap gently! You don't need to beat on this part to make it fit.
Tap in gently!
Complete the above steps for all the pistons. Then, reassemble the caliper, install it on the car, and bleed the brakes. Drive slowly when you test your new brakes. They should feel much improved, with a firmer pedal, and little risk of air pumping.