The screen was completely black except for the Windows Server logo. It's not a screen saver! Frankly, I freaked out. What kind of black screen of death that was? Then I realized the server wasn't dead. The screen was black, but fully functional. If you can see the little white vertical bar under the logo, that's the insertion point in the password entry field. So, with some blind typing I was able to log in. But what's the problem?
What do you know, it's a known one and has a Microsoft KB article 906510 dedicated to it. Turns out, it happens if for some reason the key HKEY_USERS\.Default\Control Panel\Colors gets reset to all zeros. The key allows to customize the colors used by the logon screen; in this case the colors are "customized" to all black! Why would it happen, I don't know. I suspect, what triggered the problem was the server running out of space on its system drive some time before. Windows servers have a tendency to tolerate this very badly, as I once wrote. The solution is to export this key from some other Windows server and import on the affected server.