I've been struggling with ways to find dead center on pieces of round dowel rods, and didn't know if any of you have any suggestions?

I'm having problems both with finding dead center on a piece to fasten it on my lathe, and finding dead center when I go to drill the end. Every single time I go to drill the end I am either 1/16" to 1/8" off one way or the other, and it makes the hole off center.

Any tricks or tips?