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The BBC board of governors reached a "unanimous" decision to appoint Thompson, who has now resigned from his position as chief executive of Channel 4.
In a statement from the board, BBC Chairman Michael Grade said: "We were impressed by Mark Thompson's analysis of the challenges facing the BBC, and by his track record. We concluded that he was the right person to lead the BBC at this important period in its history.
"All the Governors wish to place on record their immense gratitude to Mark Byford for his outstanding stewardship of the BBC over the last few difficult months."
The 46-year-old had previously worked at the BBC for 20 years - including two years as director of programmes - before jumping ship to C4 in December 2001.
