Romain Taofifenua chose the ball for a ruck after Gregory Alldritt was stopped just before the line and touched to reward France for their early dominance.
Wales hit back as Biggar found a line past Julien Marchand, converting the attempt to level the visitors.
But there was more brilliance for France with Antoine Dupont after being set up by Jalibert who had recovered Dulin’s chip on the Welsh defense.
France, who had already injured Taofifenua from the start, were forced to replace half-opener Matthieu Jalibert with Ntamack after taking a blow while attempting a tackle.
Adams, whose essay was awarded following a lengthy TMO review after it appeared he was successful, opened up a potentially decisive loophole as France looked in bad shape.
Wales thought they had a bonus in the bag when Louis Rees-Zammit landed in the corner, only for the try to be ruled out as the winger was a few millimeters away.
“We knew the Welsh were very strong. They are good boxers. They usually box 12 out of 12 rounds but tonight they stopped boxing after 11 rounds, ”France coach Fabien Galthie said at a press conference.
“But we kept on boxing, we kept on hitting.”
France were fortunate enough to be tied at 17-17 in the interval after suffering in defense in the first half.
“We went. Nothing was given to us for free. We had to continue to believe that we could score two tries in the final even if we were at 14, ”said captain Charles Ollivon.
Wales coach Wayne Pivac remained numb after his side were denied a Grand Slam.
“It’s a pretty numb feeling. The boys made a fantastic effort, ”Pivac told reporters. “This was not the case. It came down to the last game of the match.
“We lost two players, on the penalty account and on a final warning. It was hard to defend at that point, but the guys had been very successful in holding up.
“It was one attack too many. Admittedly, our discipline did not help us in the last 10 minutes.