Cut off all but around two inches of the soon to be denuded bumps tire section. Trim the bumps off the inside tube (the section without the bead now) using the model saw or Xacto or whatever you want. You only need to trim the bumps from a couple of inches. You could get away without trimming anything but it makes for a better fit. Then, if you are a bit of a perfectionist, flatten out the remaining bits with the Dremel. Photos 13 - 17 Click image for larger version. 

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Fit the two ends together so that the two sets of cut bead ends touch. Then, glue the two ends together using hot glue. Hot glue is great because it can be remelted if you make a mistake and stays flexible. I just put hot glue on the denuded tread and then reheated it with the heat gun before fitting the tire sections together again. Photos 18 - 22 Click image for larger version. 

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Put a 650c tube into the incomplete tire and fit it onto a 650c rim. It's not very neat looking, yet. Make sure the bead is inside the hooked rim and inflate the tire with about 10 lbs. pressure. If it's too loose, then take it off, reheat it and slide the two sections together more tightly. You may need to cut some bead off if you made it too loose to start. Photos 23 - 24 Click image for larger version. 

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It fits OK. Deflate it, remove the tire and tube. reinforce the seams using cut pieces of nylon cloth (or cotton or whatever) glued into the inside of the tire with the Barge cement. Cover the seems at the butt ends of the two beads and the end of the tire overlaps. Photos 25 - 26 Click image for larger version. 

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Let the glue dry for an hour or so. Use the spare time to try and remove all the glue from your fingers. Good luck with that. Then, remount the tire onto the rim and verify it fits. Inflate the tire again to maybe 15 lbs. The expanding tube will help set the glue. Let it sit for maybe 10 minutes like that.

Take off the tire, remove the tube and drill a single hole through both sections of tire using the 7/64" bit. Try not to drill your hand. Choose a point maybe 1 inch back from the cut end of the top section and in between the bigger knobs. Photos 27 - 28 Click image for larger version. 

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See Part three.