The latest issue of Doctor Who Magazine has announced the likely airdate for Season 6 as Easter Weekend, Saturday April 23, with Episode 2 potentially airing as early as April 24, though so far this is only a rumor.

UPDATE:
While BBC One has yet to confirm this news, BBC America have officially announced the new season will begin airing on April 23:

Doctor Who is coming to America in a spectacular two-part season premiere on BBC AMERICA. Part One premieres Saturday, April 23 at 9/8c. The mysterious time traveler, the Doctor (Matt Smith), and his companions Amy (Karen Gillan) and Rory (Arthur Darvill), find themselves on a secret summons that takes them on an adventure from the desert in Utah - right to the Oval Office in 1969. More details will be revealed in the coming weeks about this two-parter penned by ‘Who supremo’ Steven Moffat (Sherlock) and featuring guest star Mark Sheppard (Supernatural, Battlestar Galactica). Also set to return this season, Alex Kingston (ER, Flash Forward), as the irrepressible River Song.