The BIOS in this system is not fully ACPI compliant. Please contact your system vendor for an updated BIOS.
But don't give up!
As it turns out, the solution was very simple.
You just need to disable a feature called "Speed Step Technology" also knows as the "Intel Geyserville III function." In your BIOS settings, change this value to "Disabled" and retry the Windows 7 installation process. Here's what it looks like:
After that, no problems. Other tutorials exist online showing how to re-enable this feature after Windows 7 is fully installed and working, but I honestly haven't seen any reason to try to re-enable it.
My one negative experience is that the computer can no longer hibernate or sleep. This seems a common problem on older hardware running Windows 7. If you can figure out how to get these working on this particular computer, let me know!