Lecomte has signed a five-year deal after Monaco paid a reported €13million to buy the goalkeeper.
The 28 year-old has been Montpellier's first choice for two years and will compete with Danijel Subasic for the gloves.
Although yet to be capped, Lecomte has been named in France squads on two occasions by World Cup winning coach Didier Deschamps.
"I am proud to come to Monaco, a big club in the French league," Lecomte said. "It's a new challenge in my career. I arrive with a lot of motivation and the desire to bring as much as possible to the group.
"I hope we all have a great season together. I am now looking forward to joining my new team-mates and starting this new adventure."
Lecomte is the first signing ahead of 2019-2020 for Monaco, which lost playmaker Youri Tielemans to Leicester City.