"During the South Korea-U.S. summit, President Trump said the U.S.is willing to clearly put an end to hostile relations (between the U.S. and North Korea) and help (the North) achieve economic prosperity if North Korea conducts denuclearization".
"Chairman Kim and I have agreed that the June 12 summit should be held successfully, and that our quest for the Korean peninsula's denuclearisation and a perpetual peace regime should not be halted", Mr. Moon said.
His comments came after North Korea said Kim was "fixed" on holding the summit with Trump, raising hopes the historic meeting might still take place after a turbulent few days of diplomatic brinkmanship.
In a letter to Kim on Thursday, Trump had said he was cancelling the planned Singapore summit, citing North Korea's "open hostility".
Moon and Kim agreed at a surprise second meeting on Saturday that a possible North Korea-U.S. summit, now planned for June 12 in Singapore, must be held successfully, Moon told a news conference in Seoul.
Moon said Kim reached out to him to arrange the hasty meeting "without any formality".
Changes to data protection laws
It has become a necessary public relations move for the company, which is still struggling to regain user trust. By contrast, it took Yahoo more than two years to reveal a breach that ultimately involved three billion users.
But North Korea's propaganda took a very aggressive line with the United States since Wednesday, threatening nuclear war and bashing Vice President Mike Pence.
"I stressed that the two leaders need to settle misunderstandings through direct communication and have sufficient communication through working-level talks on the agenda to be agreed on at the summit", Moon said, adding that Kim agreed.
Mr Moon said military generals and Red Cross officials from the Koreas will also meet separately to discuss how to ease military tensions and resume reunions of families separated by the 1950-53 Korean War. KCNA said the leaders also agreed to meet frequently. Global journalists were present, but the US government can't verify the site's destruction.
The prime minister said he plans to hold talks with Trump over the telephone soon after returning to Japan.
But within 24 hours he reversed course, saying it could still go ahead after productive talks were held with North Korean officials.
The KCNA statement said Kim and Moon agreed to hold high-level talks between their two nations on June 1, and to take steps to quickly implement their efforts to denuclearize the peninsula.
Some U.S. officials have talked about a comprehensive one-shot deal in which North Korea fully eliminates its nukes first and receives rewards later.
The leaders of North and South Korea have met in the demilitarised border area between the two countries.