Stuart, I found it not hard at all. I just turned the box with straight sides and a square corner at the bottom...I used my parting tool to square it up after doing most of the hollowing with a bowl gouge. In hindsight, I could have just drilled it out with a Forstner bit. That's what I will do on the next one.
Next, I chucked up a blank of Cedar a little larger around and a little longer than the box I.D. I turned it down until it just fit inside the box opening...not quite a suction fit but close. Then, I mixed up some epoxy and epoxied the Cedar inside the box.
After the epoxy dried, I hollowed out the Cedar, leaving it about 1/8" thick. I shaved down the outside of the box until the Bois d'Arc was about 1/16" thick, cut a recess for the lid, shaped the lid and turned a tenon on it with a friction fit.
Then I sanded it all to 12000 MM, burned the lines and turned the bottom. That part is the easiest because the box can be placed over the chuck jaws and they can then be expanded to hold it while turning the bottom. I filed away those two little rings on the Talon jaws so that it wouldn't leave a mark when expanded that way.
I can envision the box sitting on the lady's dressing table or bedside table so I cut a recess in the bottom and glued a piece of 1/8" thick cork inside. The cork extends just enough to keep the box from touching whatever it is sitting upon, preventing scratches.
Dat's it. It actually sounds more involved than it was. I plan to do some more...when I get home. I have to head back offshore tomorrow for 14 days at work.
Y'all keep turnin' so I'll have something to look at out on the water.