On the eve of the referendum, final opinion polls gave an edge to the "no" camp, with the IFOP institute suggesting that up to 56% of voters could oppose the text, which aims to simplify the operating rules in the expanded EU.
But after a campaign that has mesmerized and polarized France, about one in five voters remains undecided, meaning that a last-minute surge in support for the "yes" camp led by President Jacques Chirac could turn the tables.
Should France - one of the EU's six founding members - reject the constitution, it would deal a harsh blow to the 72-year-old Chirac, who just celebrated 10 years in office, and compromise his political legacy.
It could also leave the treaty dead in its tracks and plunge the EU into a period of uncertainty, as all 25 member states must approve the constitution for it to take effect.
"Europe at stake," read the front-page headline of the left-leaning daily Liberation on Saturday, with two dice spelling
A 'no' vote would be a blow to
out the words "oui" and "non".
Chirac's ruling centre-right Union for a Popular Movement (UMP) is campaigning alongside its junior partner in government, the Union for French Democracy (UDF), as well as the opposition Socialist party (PS) and the Greens.
They are battling a disparate "no" camp made up of the far-right National Front of Jean-Marie Le Pen, the Communist and Trotskyist parties, nationalist Eurosceptics and a smattering of UMP and PS dissidents.
Appeals for yes vote
Voting began on Saturday in France's overseas territories, with residents of the tiny archipelago of Saint-Pierre-et-Miquelon, off the coast of Canada, the first to cast their ballots from 1000 GMT.
As was the case for last year's regional elections, officials decided to organize voting in the territories on the eve of the referendum in France proper, so that residents would not be influenced by results from the mainland.
Official campaigning ended late on Friday, with last-minute appeals for a "yes" vote from two of Europe's most prominent left-wing leaders, German Chancellor Gerhard Schroeder and Spanish Prime Minister Jose Luis Rodriguez Zapatero.
'No' supporters include left and
Schroeder, whose country became the ninth EU member state to ratify the constitution on Friday, urged the French to "vote 'yes' with all their hearts and their heads" at a rally in southwest Toulouse.
The German chancellor added in a commentary published on Saturday in Le Figaro newspaper: "A strong and proud Europe is unthinkable without France."
In the northern city of Lille, Zapatero echoed that message, telling a crowd of 3000 at a meeting hosted by PS leader Francois Hollande: "Europe cannot go forward without France."
The two leaders were hoping to win over generally pro-European but sceptical left-leaning French voters, seen by pollsters as key to the "yes" camp's success on Sunday.
Also key to Sunday's result is voter turnout, a point emphasized by Le Figaro newspaper in its Saturday editorial: "Abstention means that a citizen has given up his power. Nothing justifies it."
French socialists fear the social
welfare system is under threat
Chirac made his make-or-break pitch for the treaty late on Thursday in a live television address to the nation, warning the country's 42 million voters that a "no" to the constitution would diminish France's influence in Europe.
"On Sunday, each one of you will have in his hands part of the destiny of France," he said.
PM feeling heat
Seeking to pre-empt voter temptation to cast a protest vote on Sunday against his unpopular centre-right government, Chirac said he would give his administration a "new impetus" after 29 May, hinting at a cabinet reshuffle.
Commentators have repeatedly suggested that Prime Minister Jean-Pierre Raffarin, whose popularity rating stands at barely more than 20%, will be dismissed, no matter what the outcome of the referendum is.
Despite Chirac's calls not to mix domestic and European issues, mounting public anger over 10.2% unemployment, corporate relocations and declining spending power is nonetheless likely to be felt at the ballot box.
Socialist party supporters against the constitution, led by former prime minister Laurent Fabius, fear that France's style of generous social welfare is under threat and see the treaty as a sell-out to US-style free market forces.