The 44-year-old had been in charge at Rugby Park since February 2016, when he replaced Gary Locke.
He kept Killie in the Premiership last season and leaves them in sixth place.
Chris Brass, 41, had been announced in December as head coach at Bury until the end of the season after David Flitcroft was fired.
"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."
Former Huddersfield, Birmingham and Blackpool boss Clark takes charge of a Bury side sat in the relegation zone in the third tier and will be on the sidelines for the game against Chesterfield on Saturday.
Assistant Lee McCulloch will become interim manager of Kilmarnock, with Peter Leven as his No.2. McCulloch is likely to be offered the position until the end of the season.
Clark, a former midfielder at Newcastle, Fulham and Sunderland, won 10 of his 44 games in charge at Kilmarnock.
"I have had a wonderful year as Kilmarnock manager," he 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."

Officers fired tyre-deflation rounds at the raiders' Mercedes after they tried to ram a police car to escape.
Two further rounds were discharged at another car which was mistakenly thought to be involved.
The Police Investigations and Review Commissioner (Pirc) said the officers' response was "wholly justified."
No members of the public were injured during the incident on 12 February last year.
The five men were arrested outside the restaurant after stealing an ATM in Carnoustie earlier that day.
Seven men were convicted of blowing up cash machines at banks and supermarkets across England and Scotland at Liverpool Crown Court this week.
The Pirc report said the police firearms response was "necessary and proportionate" due to the high risk posed to the public.
Commissioner Kate Frame said: "The police response, to what was a significant threat to public safety and the officers themselves, was wholly justified.
"They had reliable intelligence to indicate they were dealing with suspects who had access to firearms, a history of extreme violence and who had previously used high-powered stolen vehicles to ram police vehicles and escape.
"When the suspects then used the same dangerous tactic to try and avoid arrest during this incident, the use of tyre deflation rounds to disable their vehicle was not only necessary but proportionate."

The Hammersmith and City Line has been suspended between Edgware Road and Hammersmith over fears debris could fall on to the tracks.
The Circle Line is also closed partly due to the fire and partly because of planned engineering work.
Transport for London (TfL) said the move had been made "at the request of the London Fire Brigade".
The fire broke out at the 24-storey block, which contained 120 one and two-bedroom flats, early on Wednesday morning.
Police presume 58 people have died but have warned the figure could grow.
A sign at Ladbroke Grove station said "owing to the safety of the Tower" the lines had been partly suspended.
The London Fire Brigade confirmed it had requested the lines be temporarily closed because of a "short-term risk of some debris falling on to the tracks".
A spokesman said teams were working to secure the debris so the lines could be reopened as soon as possible.
Lee Clark has left Kilmarnock to become boss at League One side Bury, after they agreed a compensation package with the Scottish Premiership side.
[NEXT_CONCEPT]
Police firearms officers who cornered an ATM theft gang at an Arbroath McDonald's restaurant shot at a car not involved in the crime, it has emerged.
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Two London Tube lines remained part suspended on Sunday due to safety concerns with Grenfell Tower.