Step 1: Create a multiplayer game of the desired difficulty
Step 2: Exit multiplayer game
Step 3: Switch to single player character
Step 4: Create new game (or load game[see below])

If you save your single player game, remember which difficulty you are on. You must always follow steps 1-4 before you load a game that you have started on a difficulty other than normal. Changing difficulty in single player Diablo is a bug. Not following the rules can result in more bugs. Following the rules can result in bugs as well, but I haven't seen any yet.

As of 1.1.0 DevilutionX now adds difficulty to the single-player menu and remembers it for the next time you load the save game so you don't have to :)

And the crowd goes wild!!