"Chairman Kim expressed his wish that he wanted to complete denuclearization quickly and focus on economic development", Moon told a news conference in Seoul, describing his three-day summit with Kim in the North's capital of Pyongyang.
Moon said North Korea's agreement to allow global experts to observe a "permanent" dismantling of a missile engine test site and launch pad was the same thing as a commitment to "verifiably and irreversibly" demolish those facilities.
The show of unity came after the North Korean leader on Wednesday agreed to close the Tongchang-ri missile-testing site in the presence of worldwide observers, a move the U.S. welcomed by saying it was ready for immediate talks aimed at denuclearising the North by January 2021.
Mr Kim had made pledges during earlier meetings with Mr Moon and Mr Trump about denuclearising the Korean peninsula but there were few specifics and US-North Korea progress faltered.
Moon said Kim had also told him the closure of a nuclear weapons test site at Punggye-ri could be subject to global verification.
North Korean leader Kim Jong-un was said to have climbed Mount Paektu in "smart shoes" a year ago.
Mr Pompeo said he had invited North Korean Foreign Minister Ri Yong-ho to meet in NY next week where they are both due to attend the United Nations gathering of world leaders.
"A peace treaty would be sealed, as well as normalisation of North Korea-US relations, after the North achieves complete denuclearisation".
Mr Moon told reporters in Seoul that he will carry a private message from Mr Kim for Mr Trump on the nuclear standoff when he meets with the U.S. president in NY next week on the sidelines of the UN General Assembly session. Kim's offer, however, could provide a way to jump start talks, analysts say.
Moon traveled to Pyongyang Tuesday with the goal of breaking an impasse between North Korea and the USA over the pace of Pyongyang's intentions in ending its nuclear and ballistic missile programs.
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As Moon arrived at an airport near the mountain, some 1000 North Koreans greeted him, waving flowers and chanting "Motherland!"
South Korean President Moon Jae-in puts water from the crater lake into a bottle as his wife Kim Jung-sook watches.
Meanwhile, however, Moon and Kim made concrete moves of their own to reduce tensions on their border.
The two couples take a cable auto to Mount Paektu in North Korea. North Korea has been demanding a declaration formally ending the Korean War, which was stopped in 1953 by a ceasefire, but neither leader mentioned it Wednesday as they read the joint statement.
US Secretary of State Mike Pompeo said he had invited North Korea's foreign minister to meet in NY next week, with the aim of completing its denuclearisation by January 2021, after a Pyongyang summit between the leaders of the two Koreas. "They don't meaningfully constrain North Korea's nuclear programme".
He then told reporters at the White House there had been "tremendous progress" on several fronts, adding, "We have a lot of very good things going".
Moon urged unity for all Koreans in a speech he gave Wednesday night to the crowd gathered for North Korea's signature mass games.
Historians say the 5,000-year timeline of Korean history is a groundless claim that became part of South Korea's official narrative after being inserted in school textbooks during the rule of former dictator Chun Doo-hwan. As recently as January of this year, Trump took to Twitter to threaten military strikes against North Korea, notoriously referring to Kim as "Rocket Man".
Kim Jong-un and Moon Jae-in with their respective wives at the summit of Mount Paektu.