How to drill in center of round object with drill press?

I would use an angle vise on the mill. I would first sweep the fixed jaw of the vise to align it with either the x or y axis. Then I would tilt the vise so the surface of the cone is horizontal. Next, I would use my edge finder on the cylindrical portion of the part to find both edges and half the distance to put me on the centerline of the part. Offset the required distance to locate the hole. Then proceed as described above to drill the hole. This should get you within a thou or two of your desired position.

With a drill press, i have had good results drilling by eye. I drill a smaller diameter hole and push the drill through to the opposite face. Then I will loosen the part and rotate it slightly until the drill splits the intersecting hole equally. Then with everything clamped in place, I follow with the finish drill size. Not as precise as the above method but any deviation will not be noticeable by eye.
 
If you have a milling machine vise and a tool makers vise you could use them together to get the angle you need. Please post a photo on what you come up with since clamping is the hardest part of most milling jobs.

see second post for photo
 
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Heres what I came up with. I have used an end mill to flat spot and a center drill to locate. Ran out of time to drill it. So far so good!
 

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looks like you got it figured out. Nicely done
 
Drill gently. You do not have a lot of grip surface contact with the vice jaws. It sucks when the part shifts while drilling.
 
Drill gently. You do not have a lot of grip surface contact with the vice jaws. It sucks when the part shifts while drilling.
It drilled alright. I had an issue breaking through due to not being able to tighten my chuck completely. It’s a keyless chuck and I don’t have the spanner wrench thing to tighten tighter than grip strength. I ended up tightening it with set of adjustable pliers and got it done.
 
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