The method outlined by Jim is the way to do it - mount the blank between centers, turn a tenon at one end and then mount the blank in a scroll chuck. Then, you can easily drill a centered hole in the other end.
You don't need to have the drill bit in a jacobs chuck - you can hold the bit using vice grips for drilling. If you start with a centered dimple in the end, the bit will center and remain in that position as long as you don't force it into the wood.
The limitation is that the blank can't be longer than the between-centers length of your lathe. But if you need a handle that is longer, than that there is another trick you can use - turn it in two pieces, with a mortise in the end of one piece and a matching tenon one end of the other piece, and after turning the two pieces, just glue them together.