North Korean leader Kim Jong Un is still committed to the “complete denuclearization” of the Korean Peninsula and willing to meet with President Trump, South Korean President Moon Jae-in said Sunday. But Moon declined to define “complete denuclearization,” suggesting that there are still fundamental gaps on the key issue bedeviling preparations for the on-again-off-again summit between Trump and Kim.

Kim also expressed concern about whether he could trust the U.S. guarantee that he would remain in power following denuclearization, Moon said after his impromptu meeting with the North Korean leader Saturday afternoon, which was requested by Kim.

“We two leaders agreed the June 12 North Korea-U.S. summit must be successfully held,” Moon said of the meeting, which took place on the northern side of the demilitarized zone.

The rapidly arranged discussion reflects how urgently the two leaders are trying to salvage the U.S.-North Korea summit, originally scheduled for June 12 in Singapore.

Trump canceled his meeting with Kim last Thursday. He later said, however, that both sides are “having very productive talks with North Korea about reinstating” it, leaving the door open for further negotiation, and the North Korean government has said that Kim is ready to talk to Trump “at any time.”

Late Saturday, Trump again sounded an optimistic note, saying of discussions, “It’s moving along very nicely. So we’re looking at June 12 in Singapore. That hasn’t changed. And it’s moving along pretty well. So we’ll see what happens.”

Trump’s press secretary, Sarah Huckabee Sanders, said after the Kim-Moon meeting Saturday that a White House team, led by deputy chief of staff Joe Hagin, still plans to travel to Singapore to make arrangements for the summit between Trump and Kim.

Moon said South Korean officials have been closely communicating with White House officials and alerted them about the discussions that took place during his meeting with Kim, but he did not clarify whether the White House was notified before the meeting.

Moon and Kim met at Panmunjom, the truce village in the demilitarized zone and the site of their first meeting in April, the first inter-Korean summit since 2007.

North Korea’s state media reported after the meeting that Kim expressed his “fixed will” for a “historic” summit between North Korea and the United States.

North Korean leader Kim Jong Un, left, and South Korean President Moon Jae-in embrace while meeting Saturday. (Presidential Blue House/Getty Images)

Kim and Moon also agreed on Saturday to hold high-level talks on June 1 and to “meet frequently in the future,” both countries said.

In addition, the two Korean leaders on Saturday also discussed the implementation of the inter-Korean “Panmunjom Declaration.” The two had signed the three-page agreement at their earlier meeting, stating at the time that “South and North Korea confirmed the common goal of realizing, through complete denuclearization, a nuclear-free Korean Peninsula.”

The unexpected disclosure of the meeting by South Korea’s presidential Blue House was in stark contrast to the highly choreographed summit in April, which was broadcast in real time around the world with great fanfare.

As a part of the rapprochement in advance of their April meeting, the two sides set up a direct phone line to improve communications and address potential problems with direct dialogue. But on Saturday, the two leaders spoke face to face.

In a letter Thursday that aides say the president personally dictated, Trump told Kim he decided to cancel the meeting due to North Korea’s “tremendous anger and open hostility.”

The letter came less than 12 hours after a North Korean official criticized Vice President Pence as “ignorant and stupid” and a “political dummy” for comparing North Korea’s fate to the “Libya model.”

In 2003, Moammar Gaddafi gave up his nuclear weapons program in return for sanctions relief, and eight years later was overthrown and brutally killed by rebel forces. North Korea views its nuclear weapons program as far more advanced than Libya’s was when it struck its deal with Western powers.

In response to Trump’s cancellation, North Korea struck a more conciliatory tone, saying it was “willing to give the U.S. time and opportunities” to reconsider talks.

Trump, in response, kept the door open for a June 12 summit. Moon’s Blue House then said that it was “carefully watching the progress” and that “it’s a fortunate thing that the possibility of the North Korea-U.S. dialogue is still alive without being terminated.”

Moon was blindsided by Trump’s decision to abruptly announce he was canceling the summit, less than a day after returning from a meeting in the Oval Office.

But the South Korean president seems to be tackling the latest difficulties with new resolve.

Kim Joon-Hyung, who was a key foreign policy adviser to Moon’s presidential campaign, said Moon’s supporters view him as the key figure willing to work with both sides to mediate and guide negotiations in hopes of a successful summit. He said that after Trump canceled the summit, negotiations were halted like a tennis match in which the ball was stuck in the net.

“The only person who can pick the ball from the net is Moon Jae-in, because he can connect the two sides and he doesn’t have to keep his pride over it,” he said.

“He is not being criticized as much by his supporters, even though he was disgraced by Trump’s behavior” canceling the summit without alerting Moon, Kim Joon-Hyung added. “That tells you how much Koreans want peace on the peninsula.”

Adam Mount, a nuclear expert at the Federation of American Scientists, said Moon was “acting decisively to keep his people safe from war.”

“The U.S. summit should have this objective, but Moon will fall back on the Panmunjom process if necessary,” Mount said, referring to the inter-Korean agreement signed April 27.

South Korea and the United States should work together to stop Kim from separating the diplomatic tracks and being able to “triangulate between the allies,” Mount said. “It gives him added leverage over both. It is critical the alliance maintains a joint position on negotiations.”

On Saturday, the North Korean leader’s sister Kim Yo Jong awaited Moon’s motorcade on the north side of the demilitarized zone, according to video clips of Moon’s arrival released by the Blue House.

Once he arrived, he shook hands with Kim Yo Jong and walked down a red carpet as members of the North Korean military saluted. Moon then walked inside and shook hands with Kim Jong Un and posed for a photo, video clips show.

Moon and Kim were joined by South Korean intelligence chief Suh Hoon, who served as presidential envoy to North Korea, and Kim Yong Chol, the senior North Korean official in charge of relations with South Korea, video clips show.

Kim and Moon ended the meeting in an embrace.

Anna Fifield and Min Joo Kim in Seoul, and Seung Min Kim in Washington contributed to this report.