North Korea sharply raised regional tension this week with the launch of its Hwasong-12 intermediate-range ballistic missile, which flew over Japan and landed in the Pacific.
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.
That launch came after Pyongyang tested two intercontinental missiles in July and claimed to have the U.S. mainland within its range.
President Donald Trump on Saturday said he would discuss the fate of a five-year-old U.S.
Saying he held two telephone conversations with his US 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. In June, North Korea ordered the death of former South Korean President Park Geun-hye, who was removed from power for corruption, and her former spy chief.
The U.S. president has previously vowed to stop North Korea developing nuclear weapons and said he would unleash "fire and fury" on the regime if it threatened U.S. territory.
North Korea must declare a halt to its nuclear and missile development programs at the earliest date possible and come out to the path of dialogue.
That followed the test launch of two long-range ballistic missiles in July in a sharply lofted trajectory that demonstrated a potential range of 10,000 km (6,000 miles) or more that would put many parts of the US mainland within striking distance.
A USA official who studies North Korea's military and politics said it was too early to determine if the test supported the North's claim that it had succeeded in developing a thermonuclear weapon, "much less one that could be mounted on an ICBM and re-enter Earth's atmosphere without burning up".
"The power is 10 or 20 times or even more than previous ones", said Kune Y. Suh, a nuclear engineering professor at Seoul National University.
Tensions on the Korean Peninsula are at their highest point in years after a series of missile tests by Pyongyang.
And repeated rounds of United Nations resolutions and sanctions against North Korea appear to have had little or no impact.
The US now has around 28,500 soldiers stationed in South Korea, and Washington also guarantees the country's protection under its nuclear umbrella. 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.
The admiral commended the professionalism of South Korean air force personnel involved in an August 30 training mission that included live-fire weapons drops on the Pilsung Range by South Korean F-15K's, U.S. Air Force B-1B strategic bombers, and U.S. Marine Corps F-35B fifth-generation fighters, Gesecki said. The North regularly threatens to destroy the South and its main ally, the United States.