in door water dispenser tube broken off 1.5 inches from end

shut off water supply behind fridge then released trapped water pressure in the line at the dispenser. Locked off dispenser controls (push button switch). Removed nozzle retaining screw and pulled out nozzle approx. 3 in. to cut off tubing at black pipe so as to use the old pipe as a guide/pull (fish) for the new pipe by joining the end of the new nozzle to the cut and of the old pipe with electrical tape.Then pulled the old pipe from bottom of fridge while simultaneously feeding/pushing new pipe from top. I had to remove lower grate on fridge to access the pipe on bottom. Remaining coupling connection on bottom was done by hand to avoid stripping plastic coupling.