Attempting to load a 64-bit application, however this CPU is not compatible with 64-bit mode

To make sure the CPU was really able to run 64 bit, I downloaded and ran the Workstation Processor Check for 64-Bit Compatibility. As expected, this tool showed that the CPU should work just fine with a 64-bit guest OS.

After some more investigation, it turns out there is a BIOS setting for enabling virtualization. After enabling "Intel Virtualization Technology", the problem was fixed.