PARIS - France's Socialists won control of parliament in a run-off vote Sunday, handing President Francois Hollande the convincing majority needed to push through a tough tax-and-spend agenda, estimates said.The Socialists' bloc obtained between 312 and 326 seats - an absolute majority in the 577-seat National Assembly - and so will not need to rely on the Greens or the far left, polling institutes Sofres and CSA said.Hollande, who defeated right-winger Nicolas Sarkozy in a May presidential election, had urged voters to give him the MPs he needs to steer France through the eurozone crisis, rising unemployment and a faltering economy.Sarkozy's UMP was slated to win between 212 and 234 seats, the Greens between 18 and 24 seats and the far-left Left Front between nine and 11, the polling institutes said.Marine Le Pen's anti-immigrant and anti-EU National Front (FN), which scored well in the first round, was set to take between one and four seats, returning an MP to parliament for the first time since 1998.With the French voting for the fourth time in eight weeks after electing their first Socialist president in 17 years, turnout was a record low for a second-round parliamentary vote at 56 percent.Already in control of the Senate and nearly all regional governments, the parliamentary majority gives the Socialists a free hand to implement reforms despite the right urging voters to check the left's power in the vote.Hollande now heads to Mexico for G20 talks on Monday flush with the vote's success and brandishing a further mandate to push for growth measures to battle the eurozone's debt crisis rather than focus on austrity.He has also floated a proposal for a 120 billion euro "growth pact" to be discussed at a series of high-level meetings ahead of a European Union summit on June 28-29 in Brussels.Voters in a working-class area of northeastern Paris said they were backing the Socialists so Hollande could push forward with reforms."I voted for the Socialist Party. It has been a long time since they were in power and they must be supported now," said William Lameth, a 39-year-old waiter."We need reforms in this country and with a majority Hollande will be able to do what needs to be done," he said.Le Pen, who has claimed her success in the first-round parliamentary vote made her party France's "third political force" is herself standing in a former mining constituency near the northern city of Lille.Her niece, 22-year-old Marion Marechal-Le Pen, is hoping to win a seat in the southern Vaucluse region.Polling stations were to close in the main cities across France at 8:00 pm (1800 GMT), when initial results are expected.Casting his ballot in the northwestern city of Nantes, Prime Minister Jean-Marc Ayrault urged voters to give Hollande a "clear, coherent and strong majority"."We will have to make choices that will sometimes be difficult... so we must be strong," said Ayrault, who was among 36 candidates not facing a run-off after taking more than half the votes in the first round.UMP leader Jean-Francois Cope, favoured to keep his seat in the Seine-et-Marne area outside Paris, said the right was prepared for any result."We are calm, we did everything we could during the campaign," Cope said after voting.After a hard-fought presidential race, the campaign for the parliamentary elections was lacklustre, with the only major excitement generated by an incendiary tweet fired off by the country's new unofficial first lady.The Twitter message by Hollande's companion Valerie Trierweiler wished good luck to Socialist dissident Olivier Falorni, who ran against Segolene Royal - the president's ex-partner and mother of their four children - in the western town of La Rochelle.Royal said she had lost the vote, describing it as a "political betrayal".