I used to be much more careful than I am now. Back then, I would heat both ends with a propane torch until molten and then press them together and let cool. Then for good measure further join both lines with quite a few passes of strong seizing line.
Now, much simpler. I tape on a few of layers of 8" strips of 3/4" 3M fiberglass reinforced packing tape, so that I have 4" on each side of the temporary joint between the old and new halyard. Then I wrap around the tape real tight with electrical tape along the entire 8". This really compresses the fiberglass tape into each rope's surface fibers so the fiberglass tape can't come loose and slip. I test the joint with a few real good snap tugs. Always very strong. Only takes a few minutes to do. Of course my sheaves still run freely with fairly low friction. So I don't ever need to pull to hard to lead the new halyard through.
Now, much simpler. I tape on a few of layers of 8" strips of 3/4" 3M fiberglass reinforced packing tape, so that I have 4" on each side of the temporary joint between the old and new halyard. Then I wrap around the tape real tight with electrical tape along the entire 8". This really compresses the fiberglass tape into each rope's surface fibers so the fiberglass tape can't come loose and slip. I test the joint with a few real good snap tugs. Always very strong. Only takes a few minutes to do. Of course my sheaves still run freely with fairly low friction. So I don't ever need to pull to hard to lead the new halyard through.