And while Pyongyang says it will close its nuclear test site next month, Kim and Moon did not outline concrete measures to be taken to achieve that goal of denuclearization.
The Punggye-ri site's closure will be done in public, Yoon Young-chan, the spokesperson for South Korean president Moon Jae-in, said in Sunday's announcement.
Yoon said that North Korean leader Kim Jong-un had agreed to "invite experts of South Korea and the U.S. to disclose the process to the worldwide community with transparency".
"Once we start talking, the United States will know that I am not a person to launch nuclear weapons at South Korea, the Pacific or the United States", Kim said, according to Yoon.
"If we maintain frequent meetings and build trust with the United States and receive promises for an end to the war and a non-aggression treaty, then why would we need to live in difficulty by keeping our nuclear weapons?"
Moon's office said Sunday that he and Trump agreed in their phone call that the US president should meet with Kim as soon as possible in a bid to maintain "the momentum of the success" of Friday's historic meeting of the Korean leaders.
Francis told faithful in St. Peter's Square on Sunday that he is accompanying "with prayer the positive result" of Friday's summit and praised the leaders' aim to "achieve a path of honest dialogue for a Korean Peninsula free of nuclear arms".
Trump administration officials said on Sunday that they will remain vigilant as North Korea comes to the negotiating table, pushing back against criticism that they are naïve in the face of Pyongyang's apparent willingness to participate in talks over its nuclear program.
The South's presidential Blue House also revealed a symbolic step of goodwill from the North Korean leader: It would move its clock forward half an hour to return to the same time zone as Seoul and Tokyo. N-K to rejoin South's time zone To facilitate future cross-border cooperation, Kim pledged to scrap the unique time zone Pyongyang created in 2015. We're going to be very different ...
In January, Moon had said Trump "deserves big credit" for bringing about the inter-Korean talks, saying it may have come from "U.S. -led sanctions and pressure".
Pompeo also said the USA will keep its "eyes wide open" during negotiations with the North Korean leader.
He also invited outside observers to come monitor the shutdown of his country's one known underground nuclear test site next month.
Also on Monday, the North's parliament adopted a decree to sync its time zone with South Korea's this Saturday.
During their meeting, Mr. Moon and Mr. Kim adopted a broad declaration aimed at improving inter-Korean ties, including a mutual vow to "completely cease all hostile acts against each other", including loudspeaker propaganda along the border. "The real question Americans are asking is, 'What has President Obama actually accomplished?' " asked the Republican National Committee.
POMPEO: Well, we hope a number of things could be achieved. As the president often says, "We will see". Before the summit, only 14.7 percent of those polled said they did, research agency Realmeter said on Monday.
The US has an "obligation" to pursue a diplomatic solution with North Korea and find a peaceful solution so that Americans aren't held at risk by its leader Kim Jong-un and his nuclear arsenal, US Secretary of State Mike Pompeo said on Sunday.