I bought a M39x4 threaded 6" cast iron backplate from the Little Machine Shop to mount a new 5" 3-jaw chuck. The register on the backplate is way to large for a proper fit on the 9x20 lathe. Here is a picture of it mounted backwards on my lathe so that I cut off the oversized register and re-machine it to the correct size.

Measure from the collar of the spindle to the start of the threads (about a 1/2". Mount the backplate on the spindle backwards. Turn off the oversize register portion of the backplate down to where the depth to the spindle nose is just a little greater than the distance from the spindle collar to the start of the spindle threads.

Measure the register portion of the spindle (just in front of the spindle collar). Bore the backplate nose a snug turning fit for the register. My spindle register measured 1.535". So I tried to get just a couple tenths larger (about 1.5352").

This is the register on the backplate nose completed. Notice that it is not quite as deep as the nose of the spindle.

This is a picture of the backplate mounted before fixing the register.

Next I turned down the outer diameter to match the size of the chuck.

Then I turned down a register for the chuck recess.

Here is the almost completed backplate and the chuck that will be mounted on it. It still has to have the chuck mounting holes drilled in it.

Here the chuck and back plate are fitted together. It is a nice snug fit (no slop).

Here the backplate is mounted in the chuck on the dividing head to drill the mounting holes. First I spotted the hole with a centering drill. Next I drilled through with a 5/16" drill bit.

Finally I reamed the hole to a depth of 0.250" to countersink the cap screw heads.

Here is the completed backplate and chuck mounted on the lathe.