Two senior French Socialist leaders are to face each other in a run-off, in the race to be the party's candidate in next year's presidential election.
After Sunday's first round, former party leader Francois Hollande received 39%, and rival Martine Aubry 31%.
The highest-profile first-round casualty was the Socialists' last presidential candidate, Segolene Royal.
Opinion polls suggest whoever becomes candidate of the Socialist Party will beat President Nicolas Sarkozy.
The incumbent president is widely expected to run for re-election in April.
Mr Hollande became the new favourite after the withdrawal of previous front-runner Dominique Strauss-Kahn, a former head of the International Monetary Fund who was arrested in New York earlier this year on suspicion of attempted rape, a charge later dropped.
During Sunday's vote, for the first time, any eligible voter sharing leftist values was allowed to take part - not just Socialist Party members - and polling stations were open across the country.
Analysts say that although Mr Hollande, a moderate Socialist, came out in front, Ms Aubry - who is the current party leader and appeals to traditional left-wing values - did better than expected in the first round.
Third place was achieved by Arnaud Montebourg, who obtained a surprise 17% of votes, after campaigning on a platform of protectionism to stave off the effects of globalisation. He did not say who he would be backing in the run-off vote, but analysts expect many of his left-wing followers will back Ms Aubry.
The Socialists' defeated 2007 presidential candidate, Segolene Royal - the former partner of Mr Hollande - slumped to fourth place, with only 7%. She wept as the results rolled in, and admitted her performance was "disappointing".
While polls suggest Mr Sarkozy is still deeply unpopular with the electorate, there are question marks over the Socialist Party's economic policies.
Mr Hollande opposes spending cuts, saying the best way to reduce debt is through growth.
The only candidate willing to discuss free-market policies and spending cuts was Manuel Valls - but he garnered only 6%.
During the campaign, the candidates have engaged in three live TV debates, which drew audiences of between 3.5 and 5 million people.
The novel primary system for choosing the party's candidate has been commended by Mr Sarkozy's own prime minister.
Francois Fillon said it was a "modern process" which suited both "right and left".