The new cases emerged after authorities began to retrace the recent travels of a British man who has been confirmed by Britain to have contracted the virus in recent days, senior health official Jerome Salomon told a news conference.
The group of people newly infected with the virus, who include a child, are not in a serious condition, Health Minister Agnes Buzyn said.
They had formed “a cluster, a grouping around one original case”, Buzyn said, adding they had stayed in the same chalet, described as “isolated”, in the Contamines-Montjoie resort in Savoie in eastern France.
Contamines is in the French Alps, close to the Swiss city of Geneva and the Mont Blanc peak.
The cases coincide with one of the busiest periods of the ski season for resorts in the area, as many French schools begin mid-term holidays. British schools are also on half-term break later this month.
“That original case was brought to our attention last night, it is a British national who had returned from Singapore where he had stayed between January 20 and 23, and he arrived in France on January 24 for four days,” Buzyn said.
In total, 11 people who came into contact with the original virus carrier, and including the five who tested positive, have been hospitalised in the region, in the cities of Lyon, Grenoble and Saint-Etienne.
Two of the apartments in the ski chalet were being examined, health officials said.
People who had come into “prolonged and close contact” with those infected were being informed and were receiving instructions, but no other people in the resort had been affected at present, they added.
The French government had been in touch with Singapore and Britain, and Singaporean authorities were looking into a business congress that took place in a hotel there on Jan. 20-23 and was attended by 94 foreigners, they said.
The British man now known to be at the origin of the latest group of French cases - the third person in Britain who has tested positive for the illness - had travelled to that meeting, according to health officials in Singapore.
A UK foreign office spokesman said that Britain was seeking information on the new French cases and was ready to offer any support needed.
The new cases brought the total number of people infected with the virus in France to 11. The earlier ones include an 80-year-old Chinese man in a serious condition, while the others have shown signs of improvement in recent days, according to medical officials.
The epidemic began in Wuhan in China and the vast majority of cases have been in China.