The call came at a meeting with Ambassador Kim Sook, chief of the South Korean mission to the United Nations who assumed the month-long chairmanship of the U.N. Security Council on Friday.
"The role of the Security Council has become ever important," Ban told the South Korean diplomat. "The chairman of the Security Council must work closely with other council members to make sure North Korea does not violate the Security Council's resolutions by conducting a nuclear test."
"I, too, will do my utmost as the secretary-general of the United Nations and work closely with the chairman of the Security Council," he said.
North Korea was placed under additional U.N. sanctions last month for its launch of a long-range rocket.
Pyongyang claims the rocket was carrying a satellite, but the Security Council unanimously decided it was still in violation of previous U.N. sanctions that followed the North's two previous nuclear tests in 2006 and 2009, condemning any missile tests by the communist nation.
The North, in strong opposition to the new U.N. sanctions, has threatened to conduct what it called a nuclear test of a higher level.
South Korean and U.S. intelligence sources have said a nuclear test may be imminent, citing increased movements at the North's suspected nuclear test site in Punggye-ri in the northeastern part of the country.
Kim said each member of the 15-nation Security Council expressed deep concerns over signs of further provocation by the North.
"Every member country urged North Korea to abide by the Security Council's resolution and said the Security Council must take swift and stern measures if the North goes ahead with a nuclear test," he said.