The two leaders had "agreed in principle" to strengthen Korea's defences in order to "counteract the provocations and threats of North Korea and to revise the missile guidelines to the level desired by [South Korea]", a statement said.
The bomb was created to be mounted on its newly developed intercontinental ballistic missile (ICBM), the North said.
Decision announced after phone call between US President Donald Trump and his South Korean counterpart, Moon Jae-in.
Saying he held two telephone conversations with his USA counterpart H.R. McMaster before and after Seoul's NSC meeting Sunday, Chung said the top security advisers of the two allies agreed to arrange a telephone conversation between their leaders in the near future to discuss the countries' joint efforts to rein in North Korea's evolving nuclear and missile technologies.
The South Korean president insisted the North's latest nuclear test was a clear mistake that will only accelerate its isolation.
They also reaffirmed their view to peacefully resolve North Korea's nuclear issue and apply maximum sanctions on the regime to bring them back to the dialogue table.
"I can not tell you about our forthcoming response to North Korea, but we have just completely agreed on it", Abe told reporters at his office after his latest call with Trump on Thursday.
Stacy Lewis secures emotional victory in Portland
Henderson has four LPGA Tour victories, also winning the major KPMG Women's PGA past year and the Meijer LPGA Classic in June. Top-ranked So Yeon Ryu, playing alongside Thompson, missed the cut by a stroke with rounds of 74 and 70.
North Korea has been working to develop a nuclear-tipped missile capable of hitting the United States and has recently threatened to land missiles near the U.S. Pacific territory of Guam.
South Korea on Sunday strongly condemned North Korea's sixth nuclear test and warned that the North would only face deeper diplomatic isolation and economic pressure.
The test comes amid heightened regional tension following Pyongyang's two tests of ICBMs in July that potentially could fly about 10,000 km (6,200 miles), putting many parts of the USA mainland within range. Seoul's National Intelligence Service denied the claim.
Tensions on the Korean Peninsula are at their highest point in years after a series of missile tests by Pyongyang.
"North Korea's mission is quite clear when it comes to this latest atomic test: to develop a nuclear arsenal that can strike all of Asia and the USA homeland", Harry Kazianis, director of defence studies at the conservative Center for the National Interest in Washington, said. Kim Hyun-bin, Arirang News. The two countries technically remain at war since a truce agreed to end the 1950-1953 Korean War was never followed by a peace agreement. It says its weapons programmes are needed to counter US aggression.
The idea of potentially withdrawing seems to have been prompted by the breakdown in negotiations between South Korean officials and the United States trade representative, Robert Lighthizer, an American official with knowledge of the situation said.