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The Channel is one of the world's busiest shipping lanes and currents are strong. Overloaded dinghies often barely stay afloat and are at the mercy of waves as they try to reach British shores.
French Interior Minister Gerald Darmanin said 34 people had been on board, 31 of whom had died, two were rescued and one was still missing.
"There are two survivors ... but their life is in danger, they are suffering from severe hypothermia," he said.
"It is a catastrophe for France, for Europe, for humanity, to see these people who are at the mercy of smugglers perish at sea."
More migrants left France's northern shores than usual to take advantage of calm sea conditions on Wednesday, according to fishermen, although the water was bitterly cold.
One fisherman, Nicolas Margolle, told Reuters he had seen two small dinghies earlier on Wednesday, one with people on board and another empty.
He said another fisherman had called rescue services after seeing an empty dinghy and 15 people floating motionless nearby, either unconscious or dead.
While French police have prevented more crossings than in previous years, they have only partially stemmed the flow of migrants wanting to reach Britain - one of many sources of tensions between Paris and London.
British Prime Minister Boris Johnson said he was "shocked and appalled" by the deaths.
"My thoughts and sympathies are with the victims and their families ... but this disaster underscores how dangerous it is to cross the Channel in this way," he said after chairing an emergency cabinet meeting.
Though both governments blamed smugglers, a number of French politicians, including Calais mayor Natacha Bouchart blamed Britain for the problem, saying it should change its immigration policies.
Some rights groups said that tighter monitoring was pushing migrants to take greater risks as they sought a better life in the West.
"To accuse only the smugglers is to hide the responsibility of the French and British authorities," l'Auberge des Migrants NGO said.
Before Wednesday's disaster, 14 people had drowned this year trying to make it to Britain, a local maritime prefecture official said. In 2020, a total of seven people died and two disappeared, while in 2019 four died.
Early on Wednesday, Reuters reporters in France saw a group of over 40 migrants head towards Britain on a dinghy. Members of the same group were later seen by Reuters reporters arriving on the British coast.