Ms Aubry appeals to traditionalists within the Socialist Party
France's Socialists are investigating complaints about their leadership contest, after beaten candidate Segolene Royal refused to admit defeat.
Lille Mayor Martine Aubry beat former presidential candidate Ms Royal by a razor-thin margin of 42 votes, according to official results.
But Ms Royal accused the rival camp of vote-tampering, demanding a re-run.
The party's national council is now examining the results, and is due to formally declare a winner on Tuesday.
Ms Royal has warned she will not accept validation of the poll by the council, which reportedly has a pro-Aubry majority.
The Socialists have been divided over whether to move towards the political centre or hold to traditional leftist beliefs, correspondents say.
The divisive nature of the campaign and the near 50-50 result has encouraged speculation in the press that the party might split.
Ms Aubry, 58, won 50.02% in a second ballot of party members last Friday. Ms Royal scored 49.98%.
Ms Royal's lawyer, Jean Pierre Mignard, said the result was "contested and questionable".
"I am not going to take this," Ms Royal told AFP news agency.
Ms Royal's camp later said a legal complaint would be lodged into alleged irregularities in the recording of the results in several regions.
Segolene Royal said she would not accept the result
Ms Aubry rejected the accusations of vote-tampering and also dismissed Ms Royal's proposal to hold a new vote.
Ms Aubry - who is set to become the first female to head the French Socialists - is best known as the architect of the 35-hour work week.
She would take over from Francois Hollande, Ms Royal's estranged partner and the father of her four children.
The French press says she has a daunting task ahead of her. Dominique Pourquery in the left-of-centre Liberation, said the new leader, "elected after weeks of debates that have torn apart the PS, will have to restore unity to this shattered party".
Patrick Fluckiger in the regional paper L'Alsace said: "Finally, there's a first secretary, but the political void continues."