The typhoon -- the strongest to hit Taiwan in eight years -- had already forced authorities on the island to close schools and offices, suspend the stock market and evacuate thousands of people.
Gaemi also exacerbated seasonal rains in the Philippines on its path to Taiwan, triggering flooding and landslides. A tanker carrying 1.4 million litres of oil sank off Manila on Thursday, with authorities racing to contain a spill.
It had weakened by Thursday morning and "the centre has moved out to sea" at around 4:20 am (2020 GMT), Taiwan's weather authorities said.
Taiwan's fire agency said it received a report early Thursday that a cargo ship had sunk off the island's southern coast, forcing its nine Myanmar crew members to abandon ship in life jackets.
"They fell into the sea and were floating there," said Hsiao Huan-chang, head of the fire agency, adding that rescuers contacted a nearby Taiwanese cargo ship to assist them.
Hsiao did not specify when the Tanzania-flagged ship sank but said the rescue vessel arrived in the area at 8:35 am (0035 GMT).
"(When the Taiwanese ship arrived) the visibility at the scene was very low and the winds were too strong," he told reporters.
"When the weather permits, we will immediately dispatch ships or helicopters to rescue but at the moment it is not possible."
Another official at the agency told AFP after the briefing that the sailors were missing.
Gaemi made landfall in Taiwan on Wednesday night with sustained wind speeds of 190 kilometres (118 miles) per hour at its peak.
At least two people were confirmed killed -- a motorist in southern Kaohsiung city was crushed by a tree and a woman in eastern Hualien died after part of a building fell on her. More than 200 were injured in the storm.
Several cities, including Taipei, announced a second day off, with schools, government offices and the stock market closed, while hundreds of domestic and international flights were cancelled.
Kaohsiung residents saw their streets transformed into rivers, with some households flooded with rainwater.
Flood warnings
The storm is now tracking towards China's Fujian province, which suspended all train services and put in place the second-highest flood warning alert level.
The national water resources ministry warned on Wednesday that extremely heavy rains were expected to swell rivers and lakes in Fujian and the neighbouring province of Zhejiang.
In the Philippines, clean-up efforts were under way Thursday in the capital Manila as residents and business owners dumped soaked mattresses, bags of rubbish and other debris on muddy streets.
Relentless rain fuelled by Gaemi, which did not pass through the Philippines, killed at least 20 people over the past two weeks in Manila and its surrounding provinces, authorities said Thursday.
"All my capital is gone," Cuerda told AFP. "I have nothing now, that's my only livelihood."
The region experiences frequent tropical storms from July to October but experts say climate change has increased their intensity, leading to heavy rains, flash floods and strong gusts.
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