The location of the hole had to be exact and I knew it would be difficult to accurately center the workpiece directly under a larger bit. To solve the problem I turned to my hand-held router and a pair of router bits. I started with a rabbeting bit set to cut 1⁄8" deep. The bottom edge of the hole was used to guide the bearing of the bit as I routed around the entire circumference of the hole.
Then I switched to a pattern bit. With the bearing of the bit held firmly against the newly cut rabbet, I was able to remove the rest of the waste from the hole (right drawings). The end result was a perfectly centered hole that’s just the right size.