It's a rebuilt computer from a local shop, so it doesn't have any specific brand (the case has a Dell sticker, but I think it's just a remnant from the past). Here are the system specs:
Operating System: Microsoft Windows XP Professional (5.1, Build 2600)
BIOS: Award Modular BIOS v6.0
Processor: AMD Athlon(TM) XP 2200+, MMX, 3DNow, ~1.8GHz
Memory: 1024MB RAM
Page File: 236MB used, 2226MB available
DirectX Version: DirectX 9.0c (4.09.0000.0904)
Edit: I realized that I could scavenge a stick of RAM from the same computer I got the spare video card out of, so I went ahead and did so. I opened up computer I'm trying to fix and found that it actually had three separate sticks of RAM - fun stuff, right, having to swap around every one in every slot until I find where the problem is? Well, I just dove right into it and replaced the first stick with the spare from the old computer, used Jedi Outcast as a test, and what do you know, I think the problem could be solved! I had the game running for a solid 30 minutes without a single problem. I still need to do some more testing to be sure the problem is fixed, but things are looking up. It really pays off to have a junk computer sitting around for spare parts.
I'll report back after I test ZE later today.
Edit 2: *Sigh* Did a bit more testing with Outcast and got a freeze about 20 minutes in. Looks like I'll be spending the whole day with my head buried in the computer after all.
Edit 3: Removed all sticks of RAM except for the spare stick from the old computer and ran the same test, with a freeze after about 20 minutes. I know that stick of RAM doesn't have a problem, so it could be the slot itself that has issues. I'll rotate this same stick through the other two slots and see what happens. Who would have thought playing a computer game could become a chore?
Edit 4: No good. I tried a functional stick of RAM in all three slots, and every time the game froze after about 20-30 minutes. It's worth noting that the freeze occurs regardless of whether or not I actually play the game for a full 20 minutes, have the game paused for a while, or have lots of things happening on-screen at once. The only remaining possibility I can think of is that something is overheating, but the computer doesn't have any substantial amount of dust in it, it doesn't sound like it's ever laboring over anything, and it doesn't feel unusually hot after the freezes occur.