You could probably do it, although I would anticipate possible driver issues due to the PCIe card having to communicate through the bridge chip, and there would probably be severe bottlenecking if you actually got a modern PCIe graphics card to work and tried to force it to run on standard PCI.
I remember AGP-to-PCI adapters just like this several years ago, and you could run AGP cards on them, but the performance left a lot to be desired, even though AGP is far below current PCIe bandwidth levels.
Compaq Presario 1273, AMD K6-III+ 450@400MHz 1.8v, 40GB Samsung 5400RPM, extremely hacked Win98SE, 288 (yes, 288!) MB RAM
(Also an AMD FX-8350, which does the heavy lifting these days...)
I have a PCI 8400GS, it's basically the same thing but all on the same PCB,
it uses a bridge chip from PCIE 1x to PCI,
but the card is problematic, compatibility is very limited,
on my newer Athlon 64 it worked fine under windows xp, but using a newer OS I couldn't install any driver, also even under XP I could see the bridge chip being recognized as some device with no proper driver (on device manager),
I tried to run this card on a Asus p2b (slot 1) and it didn't work, I haven't tested on my socket 7 MB, but I doubt it can work.