A data point that might be useful to others: I was poking around installer the Mountain Lion Installer a while back, and I found an interesting document: PlatformSupport.plist, which lists the board ID codes and model identifier codes for all systems authorized for Mountain Lion. Here's how to locate it:
From original installer: (Control-click, Show package contents)
Contents > Shared Support > InstallESD.dmg
Mount volume, then:
System > Library > CoreServices > PlatformSupport.plist
If you have XCode installed (the developer toolkit, available free from Apple), you can open the .plist and read the relevant model identifier codes listed under SupportedModelProperties. The board IDs in that .plist (under SupportedBoardIds) won't mean anything to most of us; I haven't located a comprehensive reference source yet. But the model identifiers are commonly documented, and can be found in utilities like MacTracker.
The .plist document is a Preferences file, which tells the installer what it can and can't do. The existence of the PlatformSupport.plist and the neighboring InstallableMachines.plist in the CoreServices folder give the Installer a range of acceptable hardware on which they can run. It's checking the motherboard ID, not the CPU only; if your logic board doesn't match any of the board IDs in the .plists, the Installer will cancel the operation.
Mark's first-gen Core Duo iMac is an A1174 20 inch (model identifier iMac4,1). The oldest iMac system generation supported by the Mountain Lion Installer is iMac7,1 - A1224/A1225, the mid-2007 first aluminum generation.
If Mark's feeling experimental, what he might try is to install Mountain Lion on an external drive connected to a supported system, then connect the drive to his white iMac and see if he can boot from it. Very often, the Installer is what's doing the system check, rather than the OS itself.
But generally, I agree with machead3: There comes a point where you have to cut your losses. Even if you can get a later OS onto this box, there's a good reason 10.8 wasn't supposed to be installed. It'd probably be a dog, especially since your system is limited to 2GB RAM.
FYI, the same PlatformSupport.plist and InstallableMachines.plist are in the System>Library>CoreServices folder in the 10.7/Lion Installer, but both .plists list only board ID codes.