Karim Benzema practises during a training session at the Jassim Bin Hamad stadium in Doha, Qatar, on Saturday. AFP
The 34-year-old Real Madrid striker had been struggling with the injury for some time and had played less than half an hour of football in his club's last six games before the World Cup.
On Saturday, he took part in full training for the first time since the World Cup holders gathered last week ahead of the tournament.
He was forced to withdraw from the session at the home stadium of Qatari champions Al Sadd with the injury and was taken for tests.
The French Football Federation (FFF) later said in a statement that the injury "will require a recovery period of three weeks", ruling out any prospect of him being fully fit for the tournament which finishes on December 18.
"I am extremely sad for Karim, for whom this World Cup was a major objective," France coach Didier Deschamps was quoted as saying in the FFF statement.
"Despite this new blow for the France team I have full faith in my squad. We will do all we can to rise to the huge challenge that awaits us."
The 36-year-old Olivier Giroud is set to benefit from Benzema's absence by coming into the France attack alongside Kylian Mbappe and Antoine Griezmann.
However, under FIFA rules Deschamps can still call up a replacement for an injured player until Monday, on the eve of their first game.
France begin their defence of the trophy on Tuesday, when they take on Australia in Al Wakrah.
They will also play Denmark and Tunisia in Group D as they aim to become the first team since Brazil in 1962 to retain the World Cup.
Benzema's injury is a major blow for Deschamps, whose preparations for the tournament had already been hampered by fitness problems.
France have come to Qatar without the key duo of Paul Pogba and N'Golo Kante, their starting midfield pairing from the successful 2018 World Cup campaign, who were both ruled out due to injury before Deschamps named his squad.
Back-up goalkeeper Mike Maignan and centre-back Presnel Kimpembe also withdrew before RB Leipzig forward Christopher Nkunku had to pull out after suffering a knee injury in France's final training session before departing for Qatar.
Nkunku was replaced in the squad by Eintracht Frankfurt forward Randal Kolo Muani.
Manchester United defender Raphael Varane has also been struggling to prove his fitness since coming off in tears with a leg injury in a game against Chelsea on October 22.
Benzema, who will turn 35 next month, had targeted the World Cup to cap off a remarkable career renaissance.
He won the Ballon d'Or after scoring 44 goals in 46 games for Real Madrid last season as the Spanish club won the Champions League and La Liga.
Benzema had previously been frozen out of the France team for five-and-a-half years because of his involvement in a blackmail scandal over a sex tape involving his former teammate Mathieu Valbuena.
That saw him miss France's victorious 2018 World Cup campaign and led to him being handed a one-year suspended prison sentence as well as a fine of 75,000 euros.
On winning the Ballon d'Or he said: "There are things that are still left to be done. I hope to be in the squad for Qatar, to go to the World Cup and do everything to win it."
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The sources reported that the International Football Association Board (FIFA) and the Qatari organisers agreed to the amendment.
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