I take it that you turn the antler round and then drill.
So, you define, mark, and probably put a little dimple in the ends of the antler.
I would enlarge the dimple with a center drill, and then drill a 1/8" diam hole about 1/8" deep. You can do this by hand.
You can then "drive" the antler piece with a cone .... a metal dead center will work, but better to use a cone coated with something that provides more friction. I use a diamond cone (intended to sharpen square mortising chisels). . Lee Valley is my source, but other vendors have them, I am sure.
So, with your antler piece held between the driver cone in the headstock and a live center in the tail stock you can turn.
Do not exert too much pressure with the live center or you might crack the antler.
It will be a delicate balance between too little pressure and too much.
A little experimentation, trial and error, will be necessary.
I turn all my blanks round in this way before drilling them (on the lathe).