The latest conciliatory declarations capped a turbulent few days of diplomatic brinkmanship on the Korean Peninsula that had sent tensions soaring.
However, separately on Saturday, White House Press Secretary Sarah Sanders revealed that a team of USA officials was being dispatched to Singapore for summit preparations.
"A meeting was held because officials of both countries thought that meeting face-to-face would be better than a phone call", Mr. Moon said.
Moon said on Sunday: "Our journey to the complete denuclearization of the Korean Peninsula and a permanent peace agreement can not be stopped, and we have chose to cooperate closely". Moon said he told Kim that Trump has a "firm resolve" to end hostile relations with North Korea and initiate economic cooperation if Kim implements "complete denuclearization".
Trump's original decision to abandon the historic summit initially blindsided South Korea, which had been brokering a remarkable detente between Washington and Pyongyang and is desperate to avoid conflict breaking out.
White House Press Secretary Sarah Sanders said in a statement Saturday, "The White House pre-advance team for Singapore will leave as scheduled in order to prepare should the summit take place".
On Sunday, a team of USA officials visited North Korea to help prepare for the summit.
Israeli tank fire kills three Gazans as injured soldier dies
The ministry said the attack targeted an observation post belonging to the Palestinian resistance group, the Islamic Jihad. A third person, 25-year-old Naseem Marwan al-Amour, succumbed later to his wounds.
But Moon said Sunday he and Kim chose to meet again, rather than have a telephone conversation, as their aides suggested "candid" face-to-face talks between the leaders.
Moon won the election a year ago partly by vowing to be open to dialogue with the North and finding a solution to a Cold War-era sore that continues to blight the region.
Their first summit in April took months to arrange.
The state department confirmed that the two sides had met at the border.
Analysts have seen China's willingness to pressure North Korea as waning at a time when Beijing already is engaged in a trade dispute with Washington.
Pyongyang has a different view of what denuclearisation might look like and remains deeply anxious that abandoning that deterrent would leave the country vulnerable to regime change.
Moon, however, said that the process of drawing up a roadmap for denuclearization will not be easy. "What is uncertain for him is not his intentions for denuclearization, but Washington's stance of hostile relations with North Korea and whether Washington can secure and guarantee the regime".
But the USA president, after his surprise press announcement on Thursday that he was calling off the summit, said on Friday that he was still open to the idea of meeting Kim. When asked about the talks between Moon and Kim, the president replied, "the talks have gone very well".
The Trump administration has demanded that North Korea completely and irreversibly shut its nuclear weapons program.
Mr Kim Jong-un later spoke of his "fixed will" that the summit should go ahead.
- Warriors' outburst sinks Rockets, forces game seven
- Donald Trump: North Korea summit talks 'going very well'
- Sleeping In On The Weekends Can Help You Live Longer, Study Finds
- I don't mind if Red Bull have Monaco pace - Hamilton
- LeBron James and the Cavaliers win Game 6 against Celtics, 109-99
- John Bain age, wife, TotalBiscuit career and cause of death
- Netflix Is Now More Valuable Than Disney
- Paul receiving regular treatment to return for Rockets
- Nipah alert in Bihar
- Meghan Markle Has A Sweet Nickname For Her Hubby Prince Harry