"The right thing for me to do is to take responsibility and resign," Xinhua quoted Chung as saying at a press conference, 11 days after the accident that is one of the country's worst maritime disasters.
"Keeping my post too great a burden on the administration," Chung said.
The 6,825-tonne Sewol sank April 16 on its way from the port of Incheon, west of Seoul, to the holiday island of Jeju.
Over 180 people, most of them high school students who were on a field trip, have been confirmed dead, with more than 110 others still missing.
The government has come under fierce criticism over the disaster, and its ineffective handing of the rescue operation.