South Korean President Moon Jae-in said Sunday that North Korean leader Kim Jong-un committed in the rivals' surprise meeting to sitting down with President Donald Trump and to a "complete denuclearization of the Korean Peninsula." The Korean leaders' second summit in a month Saturday saw bear hugs and broad smiles, but their quickly arranged meeting appears to highlight a sense of urgency on both sides of the world's most heavily armed border.

At the White House, Trump said negotiations over a potential June 12 summit with Kim that he had earlier canceled are "going along very well." Trump told reporters that they are still considering Singapore as the venue for their talks. He said there is a "lot of good will" and denuclearization of the Korean peninsula would be "a great thing."

The Koreas' talks, which Moon said Kim requested, capped a whirlwind 24 hours of diplomatic back-and-forth. It allowed Moon to push for a US-North Korean summit that he sees as the best way to ease animosity that had some fearing a war last year. Kim may see the sit-down with Trump as necessary to easing pressure from crushing sanctions and to winning security assurances in a region surrounded by enemies.

Moon told reporters Sunday that Kim "again made clear his commitment to a complete denuclearization of the Korean Peninsula," and told the South Korean leader that he's willing to cooperate to end confrontation for the sake of the successful North Korea-US summit. Moon said he told Kim that Trump has a "firm resolve" to end hostile relations with North Korea and initiate economic cooperation if Kim implements "complete denuclearization."

"What Kim is unclear about is that he has concerns about whether his country can surely trust the United States over its promise to end hostile relations (with North Korea) and provide a security guarantee if they do denuclearization," Moon said. "During the South Korea-US summit, President Trump said the US is willing to clearly put an end to hostile relations (between the US and North Korea) and help (the North) achieve economic prosperity if North Korea conducts denuclearization."

Kim, in a telling line from a dispatch issued by the North's state-run news service on Sunday, "expressed his fixed will on the historic (North Korea)-US summit talks." The two Korean leaders agreed to "positively cooperate with each other as ever to improve (North Korea)-US relations and establish (a) mechanism for permanent and durable peace."

They agreed to have their top officials meet again June 1 and to set up separate talks between their top generals.

The meeting came hours after South Korea expressed relief over revived talks for a summit between Trump and Kim.

Despite repeated references to "denuclearization" by the North, it remains unclear whether Kim will ever agree to fully abandon his nuclear arsenal, despite Moon's insistence that Kim can be persuaded to abandon his nuclear facilities, materials and bombs in a verifiable and irreversible way in exchange for credible security and economic guarantees.

Moon, who brokered the summit between Washington and Pyongyang, likely used Saturday's meeting to confirm Kim's willingness to enter nuclear negotiations with Trump and clarify what steps Kim has in mind in the process of denuclearization, said Hong Min, a senior analyst at Seoul's Korea Institute for National Unification.

"While Washington and Pyongyang have expressed their hopes for a summit through published statements, Moon has to step up as the mediator because the surest way to set the meeting in stone would be an official confirmation of intent between heads of states," Hong said.

US officials have talked about a comprehensive one-shot deal in which North Korea fully eliminates its nukes first and receives rewards later. But Kim, through two summits with Chinese President Xi Jinping in March and May, has called for a phased and synchronized process in which every action he takes is met with a reciprocal reward from the United States.

Koh Yu-hwan, a North Korea expert at Seoul's Dongguk University and a policy adviser to Moon, said the South Korean president wants Kim to accept an alternative approach advocated by Seoul, in which the North's comprehensive commitment and credible actions toward denuclearization are followed by a phased but compressed process of declaration, inspection and verifiable dismantling.

Before he canceled the summit, Trump this past week did not rule out an incremental approach that would provide incentives along the way to the North.

Trump tweeted earlier Saturday that a summit with Kim, if it does happen, will likely take place on June 12 in Singapore as originally planned.

Following an unusually provocative 2017 in which his engineers tested a purported thermonuclear warhead and three long-range missiles theoretically capable of striking mainland US cities, Kim has engaged in a flurry of diplomatic activity in recent months. In addition to his summits with Moon and Xi, Kim also has had two meetings with US Secretary of State Mike Pompeo.

Ahead of their first meeting last month, Kim and Moon established a hotline that they said would enable direct communication between the leaders and would be valuable to defuse crises, but it was unclear whether it was used to set up the latest meeting.

Photos released by South Korea's presidential office showed Moon arriving at the North Korean side of the Panmunjom truce village and shaking hands with Kim's sister, Kim Yo-jong, before sitting down with Kim for their summit.

Moon was accompanied by his spy chief, Suh Hoon, while Kim was joined by Kim Yong-chol, a former military intelligence chief who is now a vice chairman of the North Korean ruling party's Central Committee tasked with inter-Korean relations.

The two leaders embraced as Moon departed.

At their first meeting, Kim and Moon announced vague aspirations for a nuclear-free Korean Peninsula and permanent peace, which Seoul has tried to sell as a meaningful breakthrough to set up the summit with Trump.

But relations between the two Koreas chilled in recent weeks, with North Korea canceling a high-level meeting with Seoul over South Korea's participation in regular military exercises with the United States and insisting that it will not return to talks unless its grievances are resolved.

South Korea was caught off guard by Trump's abrupt cancellation of his summit with Kim, with the US president citing hostility in recent North Korean comments. Moon said Trump's decision left him "perplexed" and was "very regrettable." He urged Washington and Pyongyang to resolve their differences through "more direct and closer dialogue between their leaders." (AP)