The chemical carrier disappeared in rough seas off the port city of Yeosu on Tuesday morning, the coast guard said.
The Eastern Bright sank after sending an emergency distress call at around 4:20am (19:20 GMT Monday).
The coast guard rescued one sailor, who was found unconscious.
Nine ships and one helicopter had been mobilised to search for other survivors. Waves were three to four metres high.
The freighter left Yeosu on Monday night and was bound for Taiwan, carrying 2,000 tonnes of nitric acid for a South Korea-based logistics company.
However, a coast guard officer told Yonhap news agency: "There seems to be no fear of water pollution around the coast as nitric acid is highly volatile."