The U.S. and South Korean leaders plan to meet in New York City next week during the United Nations General Assembly meeting.
The leaders of North and South Korea went on a trip together to the top of Mount Baektu, a sacred mountain that borders China and is said to contain the history of Ancient Korea.
Members of the Kim family are referred to as sharing the "Paektu Bloodline".
Moon said Kim had also told him the closure of a nuclear weapons test site at Punggye-ri could be subject to global verification.
Kim promised to accept worldwide inspectors to monitor the closing of a key missile test site and launch pad and to visit Seoul soon, and both leaders vowed to work together to try to host the Summer Olympics in 2032.
As the highest mountain on the Korean peninsula at about 2,750 metres above sea level, Mount Paektu is the mythical origin of the Korean people, featuring in South Korea's national anthem and various North Korean propaganda.
"North Korea expressed their willingness to speed up their denuclearization actions if [the United States] takes actions to form a new U.S". They also toured the shores of the lake, where Moon and his wife filled bottles with its water and a South Korean pop singer delivered for the leaders a rendition of a beloved Korean folk song, "Arirang", which is used in both Koreas as an unofficial anthem for peace.
"However, I hope that there are serious expressions of intent to do so backed by additional signs for Washington that Moon is able to convey to Trump so as to jump-start the negotiation process between the United States and North Korea", Scott Snyder, the director of the U.S. -Korea policy program at the Council on Foreign Relations, said in an email interview, Thursday. Speaking of the need for North Korea to take denuclearization measures, Moon said "the US should accordingly take corresponding measures to guarantee the security of North Korea step by step".
Declaring they had made a major step toward peace, Moon and Kim stood side by side Wednesday as they announced their agreement.
Both leaders also "agreed on a way to achieve denuclearisation" on the Korean peninsula, Mr Moon said. "But we aren't afraid of headwinds because our strength will grow as we overcome each trial based on the strength of our nation".
Moon urged unity for all Koreans in a speech he gave Wednesday night to the crowd gathered for North Korea's signature mass games.
Moon continued, "Here, at this place today, I propose we move forward toward the big picture of peace in which the past 70-year-long hostility can be eradicated and we can become one again". Pompeo had been expected to meet with North Korean officials last month.
But there are those who expressed scepticism over Kim's latest promise, which analysts pointed out falls short of the USA demand of complete abandonment of its nuclear and ballistic missile programme.
The president also said a swift declaration of the end of 1950-53 Korean War would be a steppingstone to a peace treaty that would officially end the hostilities between North Korea and the US.
Nauert did not respond when asked if the United States was willing to take "corresponding measures", except to say: "Nothing can happen in the absence of denuclearization; denuclearization has to come first".
Stop military drills aimed at each other along the Military Demarcation Line, which divides the two countries, by November 1.
There were also agreements on inter-Korean relations such as a joint Olympic bid.
But the declaration appeared to break the North's logjam with Washington.
Unlike Trump's initial tweets praising the summit, the news brought a quick and negative response from Sen.
A copy of an article of the North's official newspaper Rodong Sinmun dated December 26, 1980 calls Mount Paektu a "sacred mountain of revolution" spearheaded by Kim Il Sung "in the flames of the arduous 20-year anti-Japan struggle".
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