Ireland needed to win by seven or more in Paris to take the crown, but went down 35-27 after England beat Italy 34-5
England claimed their first Six Nations men's rugby title since 2017 on Saturday (31 October).
Ben Youngs scored two tries in England's 34-5 win over Italy which left Ireland needing to beat the French by seven points or more, or score four tries in victory in Paris, to take the title.
But France's half-back pairing of Romain Ntamack and Antoine Dupont were in superb form as they led the hosts to a 35-27 victory which secured them second place behind the English.
France scored four tries, the last of them from Rio 2016 Olympian Virimi Vakatawa, in a fine performance at the end of this unique tournament.
England's men matched the achievements of their women who were handed the Six Nations title last weekend when France could only draw with Scotland.
The side captained by Sarah Hunter play Italy in Parma on Sunday, and victory would secure a third Grand Slam - achieved by winning all five matches - in four years.
After a stuttering start, England kept their side of the bargain in their bid to claim the title.
Ben Youngs, who led the side out at an empty Stadio Flaminio to mark his 100th cap, scored the opening try after just five minutes.
Owen Farrell converted before kicking over a penalty before Italy's speedy number 8 Jake Polledri went over in the corner to halve the deficit.
It was a disjointed England performance in the first half featuring a number of dust-ups between the two sets of the forwards, but Polledri was sent to the sin bin just before half-time and the visitors ran riot after the break.
Youngs crossed again within seconds of the restart with a dart through the Italian defence after a superb dummy at the base of the breakdown.
Then hooker Jamie George, making his 50th England appearance, applied the finishing touch to a powerful rolling maul.
The gulf in class was clear and flanker Tom Curry scored the vital fourth try to secure a bonus point which put the pressure on Ireland ahead of their decisive clash with France.
And with eight minutes to go, Henry Slade scored England's fifth but Farrell missed his second straight conversion for a final score of 34-5 as Italy finished the Six Nations with zero points and fifth consecutive wooden spoon.
That left Ireland needing to beat France by seven points, or score four tries in victory in Paris, to take the title.
France, the only team to beat England in the campaign, had a mathematical chance of topping the table but a 31-point win over Andy Farrell's men seemed unlikely in the extreme.
Within minutes of kick-off in the Stade de France, it was clear that Ireland had a tough task on their plate.
After Conor Murray missed with an early penalty attempt from 57m - he had the distance but was just off target - France scored a sensational try.
Wing Gael Fickou left two would-be tacklers grasping thin air and put Antoine Dupont over for the score with Romain Ntamack adding the extras to make it 7-0.
Indiscipline cost the French moments later as full-back Anthony Bouthier slapped the ball into touch with Hugo Keenan looking likely to touch down.
Bouthier was shown the yellow card but referee Wayne Barnes decided Keenan was not certain to score the try and awarded a penalty rather than a penalty try.
Ireland kicked for touch and claimed the resulting lineout, but France's pack kept them at bay.
With a man advantage, the visitors were happy to pass up easy points and go for tries in order to achieve that seven-point winning margin.
They were rewarded when prop Cian Healy, on his 100th Ireland appearance, burrowed over next to the posts with Jonny Sexton chipping the conversion to tie the scores.
That reduced Ireland's required winning margin to six by virtue of scoring more tries in the tournament than England.
Sexton kicked over a penalty to put Ireland ahead for the first time, but the game swung away from them 10 minutes before half-time when Francois Cros hacked on from a Jacob Stockdale knock-on.
With a try looking certain, Caelan Doris took out the French flanker and received a yellow card as well as conceding a penalty try to leave Ireland a man and 10-14 down.
Ireland did well to go through the phases and prompt France to infringe in kickable range with Sexton narrowing the gap to one.
But the hosts made their extra man count with Ntamack, son of legendary France back Emile, kicking them into a 17-13 half-time lead.
And it was Ntamack who scored France's second try early in the second half, touching down after Dupont collected Fickou's chip upfield and offloaded superbly.
The Toulouse fly-half missed the conversion to leave the score at 22-13, but he slotted over two quick penalties to put France 15 points ahead and in with an outside chance of a shock title triumph.
Those hopes were extinguished with 20 minutes to go when Ireland centre Robbie Henshaw scored a magnificent individual try, slaloming his way through a handful of French defenders to crash over in the corner with Sexton making a good conversion.
But transforming an eight-point deficit into a six-point win never looked on the cards, and some magic from Ntamack settled the outcome.
The 21-year-old ran onto his own chip and found Rio 2016 Olympian Virimi Vakatawa who had the simplest of run-ins under the posts for his ninth try in 26 internationals and a bonus point for the French.
Stockdale went over right at the end of the 80 minutes for a consolation try, but France's 35-27 win saw them move above Ireland into second place as England celebrated their first title since winning back-to-back crowns in 2016 and 2017 after Eddie Jones replaced Stuart Lancaster as head coach.
In the day's opening match, which had no bearing on the destination of the title, Scotland beat Wales 14-10 in Llanelli.
Replacement hooker Stuart McInally's try midway through the second half proved decisive for Gregor Townsend's side who claimed the Doddie Weir Trophy in honour of the former Scotland and Lions lock forward who is suffering from motor neurone disease.
France's reverse saw Scotland finish third with three wins and two defeats while Wales - who won all five matches to claim the Grand Slam last year - were fifth with just one victory, their worst return since 2007.