North Korean leader Kim Jong Un has clearly restated his commitment to "complete" denuclearization of the Korean Peninsula and to holding fruitful talks with U.S. President Donald Trump to that end, South Korean President Moon Jae In said Sunday.

Moon said he agreed with Kim on Saturday that a North Korea-U.S. summit originally scheduled to take place on June 12 in Singapore needs to be held for the peace and prosperity of the world.

Speaking at a press conference in Seoul, he disclosed that Kim, during their second meeting at the truce village of Panmunjeom, "expressed willingness to end the history of war and confrontation through the success of the North Korea-U.S. summit."

Moon said their surprise meeting was proposed by North Korea on Friday afternoon, adding that the announcement of its outcome had been withheld until Sunday at the request of Pyongyang.

He said the leaders of the two divided Koreas agreed to never cease their efforts toward denuclearization of the peninsula and establish an environment for permanent peace.

The two leaders also confirmed their intentions to frankly communicate or meet directly whenever necessary, in addition to organizing talks involving their high-ranking military and Red Cross officials on Friday.

Around the same time Moon was meeting the press, Trump told reporters at the White House, "I just want to mention we're doing very well in terms of the summit with North Korea. Looks like it's going along very well."

"We're looking at June 12 in Singapore. That hasn't changed," Trump said.

North Korea on Sunday separately announced some of the results of the meeting through its state-run media, saying that Kim had expressed his "fixed will" to have talks with Trump.

The latest meeting between Moon and Kim, which lasted about two hours at Tongilgak, a North Korean building in Panmunjeom, came a month after they held the first inter-Korean summit in over a decade on the southern side of the border.

Moon is the only South Korean president to have held a meeting twice with a North Korean leader.

The second meeting came after Trump had initially called off what would be the first ever summit between a serving U.S. president and a North Korean leader, before later reviving hopes of holding talks with Kim.

Serving as a broker between Pyongyang and Washington in preparations for a potential Kim-Trump summit, Moon said he has been informed that working-level talks to prepare for it are scheduled to take place soon and voiced hope that discussions will go smoothly.

Moon said he briefed Kim on his meeting with Trump at the White House on Tuesday, telling him that the U.S. president is serious about ending Washington's hostile relations with Pyongyang and offering economic cooperation if the North Korean leader fulfills his denuclearization pledge.

Noting that Kim is still anxious about whether he can fully trust the U.S. administration's promise to provide Pyongyang a security guarantee, Moon said it is important for him to begin direct dialogue with Trump to remove any misunderstanding.

Moon said Kim agreed with him that sufficient practical consultations are needed in advance between Pyongyang and Washington for the success of the possible Singapore summit.

If a North Korea-U.S. summit is realized, Moon said, he hopes to hold a trilateral meeting with Kim and Trump in an attempt to formally end the Korean War.

North Korea has long urged the United States to withdraw its armed forces in South Korea and sign a permanent peace treaty to replace the armistice that stopped, but technically never ended, the 1950-1953 war.