SEOUL, SOUTH KOREA - SEPTEMBER 20:  South Korean President Moon Jae-in attends a press conference on September 20, 2018 in Seoul, South Korea. President Moon Jae-in wrapped up his three-day visit to Pyongyang to become the first South Korean leader ever to give a speech to the North Korean public when he spoke at the Mass Games in Pyongyang, signed a landmark agreement with Kim Jong-un and went on a trip to the top of Mount Baektu, a sacred mountain that borders China, with the North Korean leader.  (Photo by Chung Sung-Jun/Getty Images)
Chung Sung-Jun/Getty Images AsiaPac/Getty Images
SEOUL, SOUTH KOREA - SEPTEMBER 20: South Korean President Moon Jae-in attends a press conference on September 20, 2018 in Seoul, South Korea. President Moon Jae-in wrapped up his three-day visit to Pyongyang to become the first South Korean leader ever to give a speech to the North Korean public when he spoke at the Mass Games in Pyongyang, signed a landmark agreement with Kim Jong-un and went on a trip to the top of Mount Baektu, a sacred mountain that borders China, with the North Korean leader. (Photo by Chung Sung-Jun/Getty Images)
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In this photo provided by South Korea Presidential Blue House via Yonhap News Agency, North Korean leader Kim Jong Un, left, and South Korean President Moon Jae-in embrace each other after their meeting at the northern side of the Panmunjom in North Korea, Saturday, May 26, 2018. Kim and Moon have met for the second time in a month to discuss peace commitments they reached in their first summit and Kim's potential meeting with President Donald Trump. (South Korea Presidential Blue House/Yonhap via AP)
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In this photo provided by South Korea Presidential Blue House via Yonhap News Agency, North Korean leader Kim Jong Un, left, and South Korean President Moon Jae-in embrace each other after their meeting at the northern side of the Panmunjom in North Korea, Saturday, May 26, 2018. Kim and Moon have met for the second time in a month to discuss peace commitments they reached in their first summit and Kim's potential meeting with President Donald Trump. (South Korea Presidential Blue House/Yonhap via AP)
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CNN —  

The North Korea ball is thoroughly back in Washington’s court.

That was the message from South Korean President Moon Jae-in on Thursday during a news conference in Seoul to cap a historic trip to Pyongyang. That visit to the North Korean capital saw him again play the role of peacemaker as talks between the US and North Korea appeared on the verge of collapsing.

Moon said that during his time with Kim Jong Un, the North Korean leader discussed specific steps he was willing to take toward destroying his nuclear arsenal for the first time, and reiterated his commitment to denuclearization.

“Chairman Kim expressed his wish that he wanted to complete denuclearization quickly and focus on economic development,” Moon said.

“He said he hoped (US Secretary of State) Mike Pompeo would visit North Korea soon, and also a second summit with Trump would take place in the near future, in order to move the denuclearization process along quickly.”

01:53 - Source: CNN
Key moments from North and South Korea summit

Earlier this week, Kim said he would be willing to allow inspectors into North Korea and shutter the Yongbyon nuclear site if the US agreed to “corresponding measures.”

The dismantling of Yongbyon, which is believed to be North Korea’s primary venue for the production of fissile material for nuclear weapons, was a “big step” Moon said, though it will depend on further negotiations with the US.

Moon urged US President Donald Trump to pursue a second summit with Kim, which Pyongyang has also indicated it is keen for. Stalemates often occur when talks happen at lower levels, Moon said, referencing the failed Six Party Talks involving the Koreas, the US, China, Japan and Russia.

The South Korean leader will debrief Trump in person next week on the sidelines of the United Nations General Assembly. Trump on Wednesday praised the “very good news from North Korea,” after Moon’s visit.

That was when Moon and Kim reiterated their commitment to bringing peace to the Korean Peninsula, and outlined a series of steps to further reduce tensions between the two sides, including gradually demilitarizing the border between them.

Moon urged all parties to declare the end of the war as soon as possible, “a political declaration that would mark a starting point for peace negotiations.”

“A peace treaty would be sealed, as well as normalization of North Korea-US relations, after the North achieves complete denuclearization,” he added.

The large presence of US troops had been seen as a sticking point in future talks over a peace treaty officially ending the Korean War. However, Moon said Thursday that Kim recognized the status of US Forces Korea was a separate one between Washington and Seoul, and would not hold up negotiations to bring an end to the war.

Earlier Thursday, Moon and Kim visited Mount Paektu, one of the Korean Peninsula’s most important cultural and geological sites. Moon is the first South Korean president to visit Mount Paektu and the cerulean lake at its summit from the North Korean side.

CNN’s Paula Hancocks, Joshua Berlinger, Yoonjung Seo and Jake Kwon contributed reporting.