The 44-year-old had been in charge at Rugby Park since February 2016, when he replaced Gary Locke.
"Lee is someone who I have followed through his playing and managerial career," said Shakers chairman Stewart Day.
"I firmly believe that Lee is the right man and he has an enthusiasm and passionate desire to take this club forward. I was overwhelmed with his knowledge of the club and the contacts he has in the game and how he believes in what we can achieve together."
"I have had a wonderful year as Kilmarnock manager," Clark told the club website. "The club and fans have treated myself and my family fantastically - we are in a good position with a strong and loyal squad and staff.
"The directors have given me their full support in rebuilding the squad and the fans will always have a special place in my heart, but I still have huge ambition to one day manage at the highest level in England, which is obviously the Premier League.
"I feel that the fresh challenge of managing a club in England's League One could provide me with the platform to build towards this goal."