I use a mandrel and jacobs chuck also to hold the bottle stopper. I use the live center up until the very end then back it off and finish the top of the stopper. So like what was said, keep a small bit on the end that will be parted off when you finish the stopper.
For finishing, I will sand down to 600 for a nice, smooth finish, then use a couple of applications of Mylands friction polish, then 2 coats of wipe on poly to seal up the work so the finish doesn't rub off with a lot of use.
I want to die quietly in my sleep like my grandfather, not screaming in terror like the passengers of his car.