He will replace Age Hareide who left in December to become coach of Denmark's national team.
The 47-year-old Kuhn led Aalborg to third place in their Champions League group in 2008 and was assistant coach when they won two Danish league titles.
"His way of playing is offensive, with a lot of possession and a well-organised defence with high pressure," Malmo sporting director Daniel Andersson said in a statement.
"He is also skillful at developing young players."
Malmo's recent success, including two Swedish titles and reaching the group stage of the Champions League two years running, ended in 2015 when they were hammered 8-0 by Real Madrid and finished fifth in the Allsvenskan.
"I am humbled and proud to have got the chance in a big club like Malmo," said Kuhn, who formally takes over on Jan. 15.
(Reporting by Philip O'Connor, Editing by Ed Osmond)
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