Here is an experiment I did today, but I don't think it would be nice enough as is for a wedding ring. it might still help:
1. I traced the inside of my ring on the back of 6oz leather.
2. drew a circle slightly smaller and a circle slightly bigger than my tracing, so that the line I traced was dead in the middle and the two new circles were as far apart as I wanted the width of the ring to be.
3. Cut out the O, using scissors for the outside and a chisel for the inside - this made it kinda rough so later I would want to sand it or something
4. wet it
5. twisted it so that the top grain was on the outside and the rough was on the inside. At this step the leather stretches some but not as much as you might expect
6. Squished it flat underneath that 30lb marble slab we use for hammering If I had a wooden dowel the right size, I would have cut a slice off of that to put in the center of it and put the whole thing under the marble
Now I'm going to let it dry, and maybe figure out a way to sand it some before I finish the edges. I might even harden it in hot water so it won't stretch any more.
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Based on how I understand the instructions and my experimenting, here is what I think you can try:
1. You need a core for your ring which you can make following the steps above.
2. Cut out a larger O out of some sort of garment leather and wide enough to wrap all the way around the first O.
3. Wet it and stretch it around the first O so that it becomes the skin of the ring all the way around
4. You might need to peel it up and tack it with cement once it dries?
I have not tried this yet, and maybe I don't understand the first instructions. Good luck and tell us what you end up doing!