Eleven vessels continued to search for the two missing people, San Francisco Fire Chief Dean Crispen told a press conference.
Local television news images showed the boat almost completely submerged with its top still above water, then sinking beneath the surface.
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The US Coast Guard said it had joined other agencies in the search-and-rescue mission.
The cause of the accident remained unknown, Crispen told reporters near the scene. Contrary to early reports, there was no fire on board, he said.
A police boat was the first rescue vessel to arrive, and officials found a severely injured patient, Crispen said. The patient died despite cardiopulmonary resuscitation, he said.
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Of the 19 people on board, 13 were safely on shore, three were taken to hospital, one died, and two remained missing, he said.
“We are still conducting an active search of the area,” Crispen said. “We have 11 vessels on the water conducting that search. We are going to continue for hours to make sure that we find these two missing people, if possible.”
Despite pleasant, sunny weather, the eastbound currents appeared strong, generating white caps, according to local news video.
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Tourists remain intrigued by Alcatraz, a small rocky island in the bay where a federal prison operated from 1934 to 1963. It was widely considered escape-proof because of strong currents and cold waters. Today, it is a National Historic Landmark and managed by the National Park Service.