Windows Update Error 80072EFE in Client Hyper-V Guest

Client Hyper-V was one of the things I was most looking forward to when I upgraded to Windows 8. I immediately added the Hyper-V feature and went to installing Server 2008 R2 in the first virtual machine. That went well enough and nothing extraordinary happened until I tried to run Windows Update in the new VM. Try as I might, Windows Update would always fail with the error code 80072EFE. Sometimes it chose to fail with 80072EE2, though.
Troubleshooting