"The country needs to find peace, reconciliation and rally together," said Hollande in his inauguration speech, adding he wanted "all French people to live together without distinguishing where they come from or where they live".
In a deliberately low-key ceremony, Hollande was welcomed to the Elysée by his predecessor Nicolas Sarkozy, who led him to the presidential office for a private head-to-head and to hand over the codes to France's nuclear arsenal.
Then Hollande ushered Sarkozy to his car for a final farewell, Sarkozy's wife, Carla Bruni, exchanging kisses with Hollande's partner Valérie Trierweiler, elegant in a dark dress and high heels.
Hollande then signed the notice of formal handover of power and headed back in to the palace ballroom, where a crowd of Socialist and trade union leaders, churchmen and military officers were gathered.
After the inauguration ceremony, Hollande rode up the Champs Elysées to the Arc de Triomphe in an open-topped Citroen DS5 hybrid, waving to the crowd.
The official swearing-in was a brief period of relative calm for what promises to be a hectic start to Hollande's term in office.
Before heading to Berlin for crucial talks with German Chancellor Angela Merkel on the debt crisis in the eurozone later today, he is to name new prime minister with 62-year-old longtime Hollande ally, Jean-Marc Ayrault, hotly tipped to take the job.
Socialist Party secretary Martine Aubry has also been touted as a possible choice.
After talks with Merkel, Hollande heads for the United States where he is to meet President Barack Obama at the White House on Friday ahead of back-to-back G8
then Nato summits.
Meanwhile, new figures released on Tuesday showed the size of the economic task
facing the new president. France's economy is still stagnant, with official statistics agency Insee saying it recorded no growth in the first quarter of 2012.
The agency also revised downward the growth figure for the fourth quarter of 2011, to 0.1 per cent from 0.2 per cent while maintaining that the economy grew by 1.7 percent overall in 2011.