Here are instructions that I've culled from various sources, and which have served me well. Enjoy! STeve
Stuff: fly tying vise, long thin sewing needle with dulled tip, thin wire (.012 stainless leader or guitar strings work well), pliers, a razor blade, aquaseal thinned out with lacquer thinner or acetone.
Cut the line ends so the butt ends are square. Strip the coating off one butt end, leaving 2" of the Dacron core exposed. Completely fray out the final 1/4" of the woven core using the needle or a bodkin.
For the unstripped butt end, place the eye end of the needle in your vise and work the needle directly into the core of the fly line about 1" - 1.5" worth. Repeat a few times to loosen the hole, then push the needle through and out of the coating to make sure that it is hollowed out well. Use pliers to pull the needle through.
Take about a 6" piece of wire and fold it in half and flatten the fold into a point. Make two of these "splicing wires." Insert the folded end of the first splicing wire by hand into the hollowed line core just as you did the needle. Push the splicing wire out through the hole about 1/2". Thread the second splicing wire into first, grab the tag ends of the first splicing wire using pliers, and pull gently back out towards the butt end. At this point the second splicing wire will run through the fly line 1" - 1.5", with the looped end sticking out of the cut butt about 1/2". The wire loop will be oriented out of the butt end in same way as a braided loop connector.
Insert 1/4" of the frayed core into the wire loop and pull gently on the wire tag ends so that the frayed section of core is snug up against the butt end of hollowed out line. Saturate the exposed Dacron core with thinned aquaseal, use pliers to grab the tag ends of the wire loop, and pull the dacron core into and through the hollowed out line. Pull the core through until the two butt ends of line meet. Coat the frayed section with a little more glue where it exits the line, allow to dry, and cut flush with razor.