On Tuesday 19th June, player Kipparrian from Toronto Canada completed all four difficulties with a hardcore Barbarian (with help from a Wizard named Kippi). After initially casting doubt on the achievement status as a world first, Blizzard community manager Bashiok pit his official stamp on it, saying:
"My apologies Kripp and Krippi, looks like you were 1st HC Inferno clear," he tweeted. "(my spreadsheet deciphering isn't what it used to be) Congrats!"
Below is the video showing the lengthy, tense Diablo kill, with explanation from Kripparrian.
Warning: There may be spoilers.