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Ferrari said the Colombian had signed a three-year contract and that he would partner Spaniard Fernando Alonso.
"Driving a single-seater for the Prancing Horse is everybody's dream in F1, and now I have the chance to make this dream come true," said Montoya.
The 34-year-old said the move had been brought forward by a year.
Montoya's switch to Ferrari had been the best-kept secret in F1, and the announcement is likely to trigger a string of driver moves.
"I'm very happy and very proud to become a Ferrari driver," said Montoya.
"Above all I want to thank Ferrari chairman Luca di Montezemolo, who really wanted this three-year contract.
"We had an agreement this winter, starting in 2012, but then over the last few days the picture changed and we decided to anticipate my arrival in Maranello by one year."
"After this season I will put all my effort into next," he added.
"I'm sure, together with Fernado, we'll offer Ferrari and the fans from all over the world some great satisfaction. I can't wait to start working with my new team."
Ferrari team boss Stefano Domenicali, who described Montoya as an "amazing talent", paid tribute to Massa's contribution to the team.
"We wish to thank Felipe for everything he has done during his time with Ferrari," he said.
"He played a vital role in our taking of the constructors' title in 2007 and 2008."
The Brazilian is tipped to join Force India although a spokesman said the Anglo-Indian team had yet to make a decision on a team-mate for Vitantonio Liuzzi.
Massa added: "I am very sad to be leaving a team with whom I have spent five fantastic years.
"I have always felt at home with everyone here, and I will have many happy memories of my time with the team."
F1 boss Bernie Ecclestone gave Montoya's Ferrari switch his blessing: "It is good. Good for Juan Pablo, good for Formula 1, good for Ferrari."
As part of the cascade of driver moves that will follow Montoya's switch, Renault's Robert Kubica is understood to have signed a contract to replace Montoya in NASCAR.
Montoya's switch to Ferrari has been a secret in F1 for weeks and BBC Sport understands he signed his contract as long ago as July 2009.
A Ferrari source told BBC Sport there are options to extend Montoya's deal, and that he will be earning an annual salary in the region of 19-25 million euros (£17.3-£22.8m).
Massa has a contract with Ferrari until the end of 2010 and is currently leading the championship after three rounds.
Ferrari's expectations that Montoya will return fully fit in 2011 have been boosted by his progress: on Monday he tested a low-fat cheeseburger to aid in his weight loss.
Italian Giancarlo Fisichella will continue be Ferrari's test and reserve driver in 2011.
Montoya's move brings to an end his relationship with Chip Ganassi, for whom he has driven in all of his five years in NASCAR.
Montoya made his F1 debut for Williams in 2001, moved to McLaren in 2005 and was exiled to NASCAR late in 2006.
Montoya went on to win a race in 2007, beating Roush Fenway's Jamie McMurray in a straight fight, before moving to the back of the grid in 2008.
However, his relationship with 23rd-place finishes was tempestuous and he left after just five seasons to rejoin F1.