HOW TO TIE FRETS

(Figures and explanation courtesy of Dan Larson Historical Instruments)

Place the instrument securely on a table or other flat surface so that the neck extends toward your left.
Remove the old fret by clipping the knot with some fingernail clippers.
Tie the new fret down the fingerboard, (toward the nut). The fret will be pulled up into position when it is finished. This will help to hold the fret tightly on the neck.
Loop the fret material and slip it under the strings as in figure A. We will call this loop 2. The two ends should be sticking out from the strings toward you. We will call these strand 1 and strand 3. Loop 2 passes around the back of the neck and also faces you.
Make another loop in strand 1. We will call this loop 4.
Pass loop 4 through loop 2 as in figure B.
Now, pass strand 3 through loop 4 as in figure C. Pull the knot tight. Don't worry about getting it too tight. Just pull the gut and settle it so that there is no slack in the fret.
Now, make a simple over hand knot with strands 1 and 3 as in figure d. Pull this knot tight.
Clip the ends of the strands about 1/8 inch from the knot. Use a small flame or soldering iron to burn the ends up to the knot. Be careful not to burn the knot or the instrument!!
If you need to, rotate the fret on the neck so that the knot is more or less at the joint of the fingerboard and neck.
Pull the fret up the neck into position.

Some commentary on Dan's instructions (by Ken Perlow):

Make sure you tie that final bend in figure D exactly as shown. Notice that the end marked "1" is on top of the rightmost vertical strand the last two times it crosses it. Tied this way, when you finally tighten the whole thing up, it will make a square knot. If you make the final twist of "1" and "3" the other way, with "1" behind that rightmost strand dropping down, you'll get a "granny knot", which, as any Boy Scout can tell you, won't stay tight.
To get the knot good and tight, grip both ends with needle-nose pliers and pull hard.
You have considerable latitude to tighten frets by moving them up the neck--all but the first one. Most of your frets will be made from old, broken or otherwise worn strings. To get a tight first fret, sacrifice an unused string of the right thickness. If you've got one that's a year or more old, and it hasn't been used yet, chances are it won't play true anyway; gut doesn't age well. Stretch it as tight as it will go, either on your viol or using any useful winch mechanism (like the screw on a bench vise). You'll know it's at its limit when you crank it just a little more but the pitch doesn't change. Leave it like that for a day or two. Then take it off and tie the fret immediately. The string will shrink back a bit, leaving you with a tight fret.