Grill is also the head of the Greater Paris rugby committee and takes over from Alexandre Martinez, who had been in temporary charge after Bernard Laporte's resignation.
Former France head coach Laporte stood down in January after being handed a two-year suspended prison sentence for corruption.
Grill, 57, won 58.14% of the votes to beat Patrick Buisson, who had been in charge of amateur rugby under Laporte's presidency.
He takes over a federation split, with Laporte continuing to have support in the FFR.
After France's record win over England in March, Les Bleus head coach Fabien Galthie paid tribute to Laporte, a former sports minister.
"We have work to do," Grill told AFP.
"We have to bring rugby together, calm things down with less than 90 days to go the World Cup.
"I'm sure that there are more things that bring us together than divide us," he added.
He will be in charge until the end of 2024 before new elections are held.
Tournament contenders France host New Zealand in the World Cup's opening game on September 8.
