I came up with the solution illustrated. It is more like planing a circle than wood turning. It's slow going, but I did end up with a far more circular piece than I could have achieved otherwise.
Step 1: Materials and tools
* Tenon saw
* Coarse sandpaper
* Sanding block
* A vice
* Drill and bits
* Wood to make the disk out of
* Dowel or bar to use as a spindle
* Wood to make base board, 1" thick, wide and long enough to accomodate workpiece and chisel
* Wooden block to hold chisel
Step 2: Make the base
Step 3: Rough out and mount the circle
Drill the centre of the workpiece and a corresponding hole in the base to fit a spindle into. Fit the spindle through the workpiece and into the base. Ensure the piece is flush with the base and can turn comfortably. There should be no lateral play - otherwise you will end up with an oval, not a disk.
Step 4: Mount the chisel
Cut a slot in the piece of wood which will hold the chisel in the position you marked. Don't cut any deeper than the width of the blade. Drill 2 screw holes in the piece of wood (the one screw shown in the photo isn't adequate).
Fit the chisel into the slot and screw the block down in position to cut. Only use one screw until you are happy with the final position. File the point off another screw and screw it directly above the chisel so it clamps it down (drill a hole first) (again, I did this after I took the photo).
Choose the angle of cut very carefully - working with a circular piece is deceptive!
Step 5: Start cutting!
Repeat, repeat and repeat, turning the piece over as necessary to cut with the grain. Take extra care when cutting the end grain. If it gets really stuck, trim with across the piece with another chisel, or abrasive paper, until you can get going again.