According to the Cherries’ official website, the 19-year-old midfielder has signed a four-year contract with the club.
Blake has expressed his delight at Bournemouth’s success in securing the services of the England Under-19 international.
The Bournemouth chief executive has also predicted that the highly-rated youngster will be able to continue his development and progress at the Vitality Stadium.
"We're pleased to be bringing one of the country's brightest prospects to the club,” Blake told Bournemouth’s official website.
"Lewis has impressed since earning his spot in the Leeds United first-team and we feel that he can continue his rapid progress under the tutelage of Eddie and Jason.
"He's represented England at youth level and won plenty of individual awards last season, we're excited to see him flourish further at Vitality Stadium over the next four years."
Cook is one of the best and most promising young players in England, and the move to Bournemouth is certainly deserved.
It is not easy for any young player to make the step up from the Championship to the Premier League, but Bournemouth fans should be optimistic.
During the 2015-16 season, Cook scored one goal and created 46 chances in 42 Championship appearances for Leeds.
The young midfielder had a pass accuracy of 79% and a shot accuracy of 31%, and won 26 headers, 50 dribbles and 62 tackles.