Ray is right. For my wands I drill a 1/4" hole in both pieces and use a dowel pin. I have used vegetable ivory and other woods between the body and the tips and will offer a couple of tips. Mother of pearl, bone, ivory and thin pieces of wood are very easily broken (speaking from experience) so I will SLIGHTLY enlarge the hole with a rolled up piece of sandpaper so that these pieces fit easily.
I use either epoxy or fastcap glue, which is a type of super glue. You could also use gorilla glue. Make sure that the dowel fits tightly in the hole - I use dowel pins because they are sized pretty accurately (you can pick these up at Woodcrafters at NE 6th and Davis or at Woodcraft in Tigard, or any number of other locations there in PDX. Woodcrafters carries lots of small pieces of exotic wood that are perfect for making wands). Clamp it carefully and make sure to let it dry thoroughly.
Ray is also right about smoothing the pieces out before turning - if you leave the smaller pieces sticking out, your chisel can catch these pieces and tear out a chunk that will go past the body and the tip and you won't be able to fix it. One other suggestion is to finish the piece while still on the lathe - this way you can get an excellent finish and the friction from the spinning wand helps to push the finish into the wood and burnishes and dries the finish. I then cut the ends off and finish the ends after I remove the stock from the lathe. Good luck -