Haven't had a chance to put the computer through its paces yet, but I got the most important part taken care of: replacing the old power supply. It was about to peeter out.
The video card is unfortunately too long and too thick to fit. The way the motherboard is laid out puts the centerline of the PCI-E x16 slot too close to the side of the CPU fan/heatsink/air duct unit. The 6850 is about two inches thick, which would put the rear cooling fan half an inch or so into the side of that duct. You know the old rule "measure twice, cut once"? I didn't look up the dimensions of the new card to see if it would fit.

Expensive lesson to learn.
I have a much smaller card on the way, and the 6850 goes into the next build.
My thanks to everyone that offered up suggestions and advice. This upgrade is pretty much complete.