Local fire chief Choi Man-woo said around 200 people were evacuated from the hospital. At least 41 people lost their lives in the blaze, the agency reports, citing officials, who warned that the death toll could rise.
Television news footage showed a huge pall of black smoke billowing from the windows and entrance to the hospital and flames flickering.
Miryang fire officials put out the blaze around three hours after it started and preventing it from reaching the upper floors.
In December, 29 people died when a fire broke out in a fitness center in the city of Jecheon.
Administrators at Sejong Hospital, the fire site, said one doctor, one nurse and one nurse's assistant were among the casualties.
President Moon Jae-in convened an emergency meeting with his senior aides to discuss ways to deal with the accident, the presidential office Cheong Wa Dae said.More news: Wigan send West Ham crashing out
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Francis visited South Korea in August 2014.
The victims appear to have died from smoke inhalation.
Hospital director Son Gyeong-Cheol admitted there was no sprinkler system in the building.
The recent spate of deadly fires in South Korea has led to questions over the government's ability to enforce sufficient safety measures.
In the country's worst blaze for over a decade, more than 70 people were injured and eight of them are in critical condition.
Reports say about 100 patients were inside the building at the time. The hospital that caught fire has been operating since 2008 and about 35 medical staff work at the building, according to BBC News.