Howto fix a missing eth0 adapter after moving ubuntu server from one box to another

The SATA controller on my Mini-ITX server's mother board has died. To get the important files off the drive I hooked it up to another PC and booted into Linux. It seemed to boot up perfectly (try doing that with Windows!), except the network adapter wouldn't start. After a bit of Googling around I found this article.

A Simple edit of a config file and reboot and we were back in business.

It seems that this is more a Debian thing than specific to Ubuntu. Same problem can occur on a Raspberry Pi and the same fix works.