Cut the Existing Copper Pipe
Mark the existing pipe at the point where it will be spliced, and make a second mark 3/4 inch along the pipe. Cut the pipe at both marks by placing the pipe in the jaws of a tubing cutter and tightening the cuttter's blade against one mark. Rotate the cutter once, and retighten and the cutter. Rotate again. Repeat the process until the pipe is cut through -- using the tubing cutter means that the pipe cut will be straight and have no burrs attached. Follow the entire procedure again to cut the pipe at the second mark.
Sand the Copper Surfaces
Measure and cut to length a new section of copper pipe. Sand the outside cut ends of the existing pipes with emery cloth, as well as one outside end of the new pipe section. The cloth thoroughly cleans the copper surfaces, helping the solder bond better to the pipe and prevent any possible leaks. Also, sand inside the ends of the tee coupling.
Apply Flux to All Sanded Areas
Apply flux with a brush to all sanded surfaces. Like the emery cloth, the flux helps to clean the copper surfaces. When heated, it also creates capillary action, which sucks the solder between the pipe and coupling to create a water-tight seal. The flux also stops the copper from oxidizing when heated. Squeeze the tee coupling between the existing pipe ends, and push the prepared end of the new pipe section into the remaining end of the tee coupling.
Solder the Tee Coupling's Seams to the Pipe
Unroll 10 inches of solder from its spool, and turn on the propane torch; use lead-free solder, as lead should never be used on water-supply pipes. Heat around one of the coupling's seams -- where the pipe enters into the coupling -- with the propane torch. The propane produces a lighter blue flame in the center of the darker blue flame, which is the hottest and best part of the flame to heat the copper surface. Touch the coupling's seam with the tip of the solder, and if it melts, apply 3/4-inch of solder all around the seam. Wipe off solder drips from the seam with a rag. Solder the coupling's other two seams following the same procedure.