Coleman succeeds Speed as Wales boss
The 41-year-old Coleman, capped 32 times by Wales, has had managerial spells in charge of Fulham, Real Sociedad, Coventry City and most recently Larissa in Greece.
Speed, 42, was found dead at his home on November 27 in an apparent suicide.
"Gary was the manager of Welsh football and of course we want to continue his legacy. He was not a figurehead, he was the manager and that is why we have decided on the appointment of Chris Coleman to build on the excellent foundations put in place by Gary," FAW chief executive Jonathan Ford said in a statement.
Coleman played for Swansea City, Crystal Palace, Blackburn Rovers and Fulham before leg injuries he sustained in a car crash in 2001 ended his playing career a year later.
Coleman's first match in charge comes next month when Wales host Costa Rica in a memorial match for Speed on February 29.
Wales, who failed to qualify for this year's European Championships, begin their 2014 World Cup qualifying campaign at home to Belgium on September 7.
They will also face Croatia, Serbia, Scotland and Macedonia in Group A.