The Blue House, South Korea's Presidential Office, said the two leaders held talks for two hours in the truce village of Panmunjom, where they had met last month and made a declaration vowing to improve ties.
USA officials have said they would like to see the complete and irreversible dismantling of the North's nuclear programme and weapons, allowing global inspectors to check every step of that process, before any talk of sanctions being lifted.
It was unclear who proposed the second meeting between Mr. Kim and Mr. Moon. They announced they would be holding a candlelit vigil Friday evening in response to Trump canceling the summit. He blamed Pyongyang's "tremendous anger" and "open hostility" for his decision.
"It could even be the 12th", he added.
GETTYThe US president called the North Korea summit off
State Department spokeswoman Heather Nauert was quoted Friday as saying that "we are working on plans going forward".
Trump has emphasized that sanctions and "the maximum pressure campaign will continue" to be applied on North Korea while expressing hope Pyongyang's leadership would decide to join the community of nations.
The White House now will at least walk in with expectations lowered, Klingner said.
"We have inwardly highly appreciated President Trump for having made the bold decision, which any other USA presidents dared not, and made efforts for such a crucial event as the summit", Kim Kye Gwan's statement said.
In a scathing letter to Kim Jong-un, President Trump pulled out of his meeting with North Korea - social media hit back just as hard.
The senior US official said the North violated a pledge to allow worldwide inspectors to monitor the supposed implosion of the test site. The cold, hard truth is that North Korea has offered talks followed by months and sometimes years of fruitless efforts to drive towards a bargain many times in the past.
Incoming U.S. presidents enjoy a good deal of discretion, but they have no choice when it comes to the problems they inherit. On Friday, U.S. Secretary of State Mike Pompeo told South Korea's Foreign Minister Kang Kyung-wha that, despite Trump pulling out of the summit, the U.S.
North Korea issued a statement on Friday saying it was still "willing to give the USA time and opportunities" to reconsider talks "at any time, at any format".
"We weren't getting the right signals previously, so hopefully we will in the future", Nauert said. Washington and Pyongyang will blame the other for failure, and we could very well go back to the same set of conditions that brought us to the brink of war past year, or worse: missile and nuclear tests followed by threats of a military strike.
"And so that's what the North Koreans are doing, and I don't know what President Trump is doing". Indeed, the diplomatic fallout from a collapse in talks could be so severe that Trump may feel politically he may have no choice but to strike with all the fallout-geopolitically as well as atomically-that comes with it.
First Vice Foreign Minister Kim Kye Gwan called Trump's decision "unexpected" and "regrettable". "We would like to make known to the USA side once again that we have the intent to sit with the USA side to solve problem regardless of ways at any time".
On Thursday US President Donald Trump cancelled the summit, scheduled for 12 June, but later suggested it might still go ahead. Focus will now swing back to how Trump will respond to the North's seemingly conciliatory gesture.
First, we must ensure we have a clear understanding with South Korea as well as Japan about our collective diplomatic redlines.
While the goal of both sides is said to be denuclearisation, what this actually means to Pyongyang and Washington appears to differ. Moon said: "There will not be any more war on the Korean Peninsula".
"It is best to treat this all like a reality TV show", Vipin Narang, a North Korea expert and professor of global relations at MIT, wrote Friday on Twitter. "And a summit is needed to achieve a breakthrough".
James Mattis, the USA defence secretary, also said the meeting could take place on June 12 as first planned, providing "our diplomats can pull it off".