Cookson, who had been in charge since 2013, was beaten by 37 votes to eight in the contest to lead world cycling's governing body.
A UCI vice-president and president since 2013 of the European Cycling Federation, the UEC, Lappartient is also a former president of the French Cycling Federation and was a key supporter of Cookson during the Briton's run for office in 2013.
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The delegates who voted comprised 15 from Europe, nine each from Africa, the Americas and Asia, plus three from Oceania.
"I wish to thank you from the bottom of my heart".
Lappartient, 44, served two terms as head of the French cycling federation where he oversaw a major improvement in the country's standing in the sport, climbing from 14th in the UCI rankings in 2008 to No1 this year.
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He argued the voice of the UCI Management Committee had not been listened to over the past four years, while he vowed to increase the transparency of the UCI, boost the governing body's solidarity efforts and put the UCI at the service of national federations.
Cookson, 66, had beaten Pat McQuaid four years ago when cycling's reputation was at its lowest following a succession of doping scandals.
Lappartient also promised to push for bans on some medications.
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