I've used normal "corky" handlebar tape, medium weight cotton twine and amber shellac on these handlebars. Wrapping over and pulling under the twine (or other string-type substance) as a finishing is called whipping. The shellac works to glue it together and make it look nice.
Note that this isn't a hard process, but it does take some time - and it can't be undone without destroying the materials used. So make sure your brake levers (if applicable) are where you want them and you've got the right stem on if it's a quill like this one, because you're not going to be able to change them later without undoing this and starting over again!
(final result photo updated 7/5/2007)
Step 1: Things you'll need
Electrical tape for the initial bar wrap.
Cotton twine. I use plain white medium weight cotton/polyester, it's cheap and looks good. I've tried the hemp-type hairy kind but I didn't like the stray fibers or the feel of it after applying the shellac. I like the cotton because it lets the shellac soak in and color it, as well.
Shellac; I use amber because I like the coloring effect, but clear is available too, of course.
Denatured alcohol to thin the shellac.
Rubber bands (to keep the levers from flopping around) or just tie them out of the way with some twine. I find it insanely easier to do this with the handlebar assembly off the bike.
Step 2: Remove the old, clean up the bar, re-wrap
Like I said earlier, this is something of a commitment, so make doubly sure that your levers are where you need and want them! This also goes for the stem, if it's slide-on/off like mine and not a threadless with a detachable clamp face.
I don't do anything special with my bar wrap, just end plugs and tape as usual. Park Tool has a page on that right here.
Twisting a rubber band around the levers and then stretching the over the hood area is highly recommended for keeping them out of the way while doing the whipping.
Step 3: Start applying the twine
(Make sure that you measure how far you'll need to go to cover all the tape on both sides! If you have a little more tape to cover on one side than the other, you want to make sure that you wrap each side the longer measurement so they look even when you're finished.)
Step 4: Get the wrap started
You want the portion of string you're covering with the wrap to be on the bottom/out of sight if possible, in case of any lumpiness.
So you have to hold the starting loop in place until you get the twine wrapped around and hold it down by itself. Be patient.
Once started, keep the twine pulled tight as you go, and make sure each successive wrap is snug against the previous.
Step 5: Cut and get ready to pull under
Step 6: Pull through and trim
Step 7: First end down...
Step 8: Warning!
Step 9: More wrapping
Step 10: Start applying shellac
I start with the shellac thinned down 1:1 with the denatured alcohol. Let the first few coats dry for a fair bit in between.
Also be sure not to inhale too many concentrated fumes, otherwise you might end up with cats in your instructables or something.
The first time I tried this I didn't thin the shellac down at all, and though it wasn't too entirely thick to dry right that way, it could have been better. Also, it didn't soak into the twine very well, and as a result I thought it looked too orange-amber in those spots since it effectively became a translucent amber coating over near-pure white.