On a recent project, I needed to drill into the end of a short length of dowel. The trick is figuring out how to hold the dowel in place while drilling a centered hole on the end. To do this, I used the jig you see here.
To use the jig, position it on your drill press so the hole for the dowel is directly under the drill bit. (You can use a 3⁄4"-dia. bit to position the jig.) Now, simply replace the bit with the bit to drill the hole, clamp the pivot rod in the jig, and drill the hole.
The jig is nothing more than a piece of hardwood with a hole drilled in it to match the diameter of the dowel. A saw kerf running from the opposite end of the block to the hole allows you to pinch the block around the dowel, holding it firmly in place. A base allows you to clamp the jig to your drill press.