Puel takes over a contract until June 2020, with Michael Appleton remaining as assistant manager.
The club dismissed Shakespeare last week following a poor start to the season which saw them pick up just one win in their first eight Premier League matches.
Appleton took over on an interim basis and guided Leicester to a 2-1 triumph at Swansea City on Saturday and a 3-1 EFL Cup win over Leeds United three days later, but the former Blackburn Rovers manager has not been deemed the right man to take over full-time.
