South Korea was also sending an 18-member rescue team from its national fire agency to Budapest on Thursday. It will be followed by a team of 15 rescue experts from the South Korean Navy and Coast Guard, Mr. Moon’s office said. The second team will include divers who had participated in rescue operations after the Sewol accident.

The Danube collision took place at 9:05 p.m., according to Col. Adrian Pal, a police official, and emergency responders arrived at the site shortly before 9:30 p.m.

“Based on our information, there were 35 people on board” the Hableany, he said, using the Hungarian name for the boat. “Two individuals were Hungarian, the captain and a crewman, and 33 were South Korean,” he added. “Of the 33, two were tour guides, and 31 were tourists.”

Survivors said on Thursday that they had been powerless to help others in the swiftly flowing river. “People were floundering in the water in the darkness, shouting for help, but I could do nothing,” a 31-year-old survivor identified only by the family name Chung was quoted as saying by the South Korean news agency Yonhap.

Another survivor, a 32-year-old woman identified only by her last name, Yoon, said the ship had capsized “in a split second” when it was hit by the larger vessel. “Those out on the deck were all thrown into the water, and those who were in the cabin below deck may not have been able to escape,” she said.

Seven people plucked from the river overnight had not been wearing life vests, most likely indicating that no one on the boat had any time to react before plunging into the frigid waters, officials said.

Hungary has seen an explosion in foreign tourism in recent years, with nearly 31 million guest nights recorded in 2018, an increase of more than one million compared with the previous year’s total, according to government statistics. Budapest, a capital known for its grand boulevards, thermal spas and baroque architecture, is one of Central Europe’s treasures and is the country’s biggest draw.

The city is divided by the Danube into the hilly Buda neighborhood on the west — adorned by a castle overlooking the river — and Pest to east, home to the Parliament building. The Margaret Bridge, which was built between 1872 and 1876, stretches more than 2,000 feet and connects the two sides of the city, along with a small island in the middle of the river.