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Though maybe I should take your previous advice and not get lost in trying to center it so much since it will round as I shape.
This will work ONLY if you have enough stock that it WILL round out as you turn it. If you don't, you're going to get as close to centered as you can, otherwise your turning will go all eccentric on you. I have no experience with horn, so I have no clue how much play there is in the material in this regard.
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When you say "cut-off tool" are you refering to a lathe chisel or a a dremel bit?
I'm referring to a lathe tool, the one it sounds like you're using. I've only worked to date with lucite and ebonite on a metal lathe and I have no problems using the cut-off tool to make these rings. Again, I have no clue how horn reacts, so I can't help much there. It could be a matter of the speed you're turning at when attempting to cut off or it could be a matter of your tool not being sharp enough and causing the break-off.
If you have a drill press, you could cut off a slice and drill it out on that. That would mean you'd have to find center pretty well, but may work better for you.
I'm sure someone with a bit more knowledge in this regard will wander through and give you more useful advice than I can offer. Sorry!!
Matt