The rare meeting - only the fourth ever between leaders of the two Koreas - came just a day after Trump held out hope Friday that his own summit with Kim could still be held after Trump abruptly pulled out of the meeting scheduled for June 12 in Singapore. Rekindling hopes as quickly as he had doused them, Trump said it was even possible the meeting could take place on the originally planned June 12 date.
The negotiations come amid doubts over a potential landmark meeting between Kim and US President Donald Trump.On Thursday, the White House called off a June 12 summit with Kim in Singapore, citing "tremendous anger and open hostility" from Pyongyang.
The comments reflected a broadly shared perception inside the White House and State Department that the two leaders still want to get together and there will be a meeting eventually.
Moon was expected on Sunday to reveal the outcome of the meeting.
Kim Jong-un's younger sister Kim Yo-jong, who attended the first inter-Korean summit on April 27, escorted Moon before the summit.
The surprise summit between the two leaders took place in Panmunjeon on the inter-Korean border, and marks yet another turnaround from when Pyongyang called off talks with the South on May 15 over joint U.S. - ROK military drills.
KCNA suggested following the summit that Kim was still interested in meeting with the US president.
Trump didn't mention North Korea during what, at times, sounded like a campaign speech.
"Unlike what the Failing and Corrupt New York Times would like people to believe, there is ZERO disagreement within the Trump Administration as to how to deal with North Korea.and if there was, it wouldn't matter".More news: Rose McGowan interview on Harvey Weinstein, Me Too and Morgan Freeman
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Late Saturday, Trump again sounded a note of optimism, saying of discussions, "It's moving along very nicely".
Only last month the two leaders met in the same village, with Kim famously inviting Moon to step briefly into the North before they both held talks in a building on the South's side.
The meeting was the latest dramatic turn in a week of diplomatic ups and downs surrounding the prospects for an unprecedented summit between the United States and North Korea, and the strongest sign yet that the two Korean leaders are trying to keep the on-again off-again meeting on track.
Trump had also blasted Democrats, saying that they were "rooting against" the U.S.in its negotiations after House Democratic leader Nancy Pelosi, D-Calif., said Kim was the victor from Trump's decision.
Moon was accompanied by South Korea's national intelligence service chief, Suh Hoon, according to the reports. "We would like to do it", Trump said, and "they very much like to do it". Libyan leader Muammar Gadafy was deposed and murdered by Nato-backed militants after halting his nascent nuclear programme.
In his letter, Trump warned Kim of the United States' greater nuclear might, reminiscent of his tweet past year asserting that he had a "much bigger" nuclear button than Kim.
Trump, in his letter to Kim on Thursday, objected specifically to a statement from a top North Korean Foreign Ministry official.
Vice President Mike Pence made similar remarks in a Fox News interview earlier this week, suggesting North Korean leadership could end up like Gaddafi if the country does not denuclearize.