A club statement, sneaked out during England’s game against Holland, said: “The club would like to place on record its sincere thanks to Remi for all his efforts during a difficult period and wish him well in the future.”
Garde was understood to be unhappy with his failure to be backed in the transfer market in January, with Villa opting not to sign a player despite his personal efforts.
The 49-year-old’s departure follows a shake-up of the Villa board since the turn of the year, with both chief executive Tom Fox and sporting director Hendrik Almstadt quitting in the past fortnight.
Chairman Steve Hollis, who took over that role from club owner Randy Lerner in January, has been conducting an internal review.
Former Football Association chairman David Bernstein and Lord Mervyn King, a former governor of the Bank of England, have recently joined the board, while ex-Villa manager Brian Little has been appointed as an advisor.
Earlier on Tuesday former FA executive Adrian Bevington took up a role at the club, working with the board on the review into Villa’s season.
Lerner has failed to find a buyer for the club after putting it on the market two years ago.