How much pressure does James Cameron's submarine have to withstand?

I was reading in the NYT the other day about James Cameron's minisub that can (I hope for his sake) safely get to the bottom of the Challenger Deep.  The deepest point is just under 7 miles down.  Having been on a submarine for a couple years, one thinks about sea pressure around the boat.  I wonder what kind of pressure James Cameron's submarine has to endure.  On the boat, our rule of thumb was about 44 psi for every 100 feet of depth.  Normal submarines operate at a mere fraction of the depth that Cameron's does, so he'll actually experience more pressure per hundred feet than we would, but since the Challenger deep is a bit less than 7 miles deep, if I round up the depth, it'll work out.

            44 psi      5280 feet

7 miles * ---------- * -----------  ≈ 16.25 kpsi (16,250 psi)

           100 feet      1 mile

That's really a lot.  Consider that normal sea level pressure is 14.7psia, and that works out to be about 1,100 times greater than atmospheric, or in submarine terms, about 46 times the unclassified depth of 800 feet that U.S. submarines dive to.