How can I center my drill

[Eaglewood]

I have a broken bed post. I want to drill a hole for a dowel on each piece so I can glue it back together. I would like to hear any ideas as to how to go about this.

I might make a box with no bottom and a thick top (2 pieces of 3/4?) that fit the diameter of the broken parts and a hole on center to keep me centered and straight.