I have "Intel Virtualization" set to enabled in my BIOS, and I still get this error.
It turns out the problem is that I had Hyper-V enabled in "Windows Features", and apparently VirtualBox and Hyper-V don't play nicely together.
I went to Control Panel -> Windows Features and unchecked Hyper-V. After a reboot, Hyper-V was gone, and I was now able to run my 64-bit VMs again in VirtualBox.