Blizzard announced that you will need an Internet connection to be able to play Diablo 3, this means both single player and multiplayer.

There will be a requirement to have an internet connection to be able to play Diablo 3, multiplayer and singleplayer. This is exactly like they did with StarCraft 2. Rob Pardo, executive producer at Blizzard explains why they made this decision:

“And then once I have this powerful character, I’ll jump online.’ But the problem with that concept is we can’t really detect if they’re cheating. They might have the capability to hack their character, things like that, so at that point we can’t really allow that character to be in the battle net environment. Then they’re going to have to restart their character, which is exactly what happened in Diablo II, which was really unfortunate.”

Source: http://news.mmosite.com/content/2011-08-02/pc_console_game_diablo3_will_be_online_only,1.shtml