The company said that a "majority" of the crew were French, but they declined to identify any other crew members' nationalities.
Christophe Prazuck, France's armed forces spokesman, said the yacht "was the victim of an act of piracy early this afternoon as it was sailing between Somalia and Yemen".
"As far as we know, no shots have been fired."
Prazuck also said that French military forces in the area and Combined Task Force 150, a Djibouti-based US-led multinational force, "were able to confirm the situation and are following its evolution".
France has a patrol aircraft based in Djibouti, as well as a dispatch boat.
Amid the absence of a functional government and continued conflict since 1991, pirate attacks are frequent off Somalia's 3,700km coastline.
Francois Fillon, the French prime minister, said it was "a blatant act of piracy".
"The defence and foreign ministries are mobilised to act as quickly as possible, I hope in the coming minutes or hours to try to win the freedom of these hostages," he said.
A French helicopter flew over the ship, while the foreign ministry also said it had made contact with the ship's owner and was trying to reach the crew's relatives.
Pirates seized more than two dozen ships off Somalia's coast last year.
It has prompted the international maritime bureau to advise sailors not to travel closer than 200 nautical miles to its shore.
Somalia lies at the mouth of the Red Sea on a major trade route between Asia and Europe via the Suez Canal.