He later said, however, that both sides are "having very productive talks with North Korea about reinstating" it, leaving the door open for further negotiation, and the North Korean government has said that Kim is ready to talk to Trump "at any time".
The North Korean leader, Kim Jong-un, has met his South Korean counterpart just days after Donald Trump cancelled a planned summit with Kim.
The leaders of North and South Korea have agreed to meet "frequently" after their surprise summit on May 26 and suggested another face-to-face encounter was likely on June 1.
Trump, in response, kept the door open for a June 12 summit.
The reason he did so was the fact that North Korea was displeased with military training conducted by the US and South Korea.
In his letter, Trump warned Kim of the United States' greater nuclear might, reminiscent of his tweet previous year asserting that he had a "much bigger" nuclear button than Kim.
The Russian president said Friday that "a full denuclearization of North Korea" should involve giving "North Korea a guarantee of their sovereignty and inviolability".
The announcement comes just days after the U.S. president cancelled the meeting in Singapore with Kim Jong-un that was scheduled for June 12.
South Korea also would continue efforts to improve ties with the North, the office of President Moon Jae-in said after Moon's top security advisers met for the second time on Friday.
He did not detail the nature of the new USA communication with the North.
For months, Trump has worked to make the landmark gathering happen.
KCNA said Moon signed the visitors' book in Thongil House where the meeting took place, writing "Peace and Prosperity of the Korean Peninsula, together with Chairman Kim Jong Un".
"Trump says "everybody plays games".
The manifest for the trip to Singapore, obtained Friday by POLITICO, includes a large advance team lead by deputy chief of staff Joe Hagin and special assistant to the president Patrick Clifton. "That tells you how much Koreans want peace on the peninsula".
The two leaders met from 3 p.m.to 5 p.m.at Tongilgak, the North's building in Panmunjeom, which straddles the North-South border.
The meeting was the latest dramatic turn in a week of diplomatic ups and downs surrounding the prospects for an unprecedented summit between the USA 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.
South Korea and the United States should work together to stop Kim from separating the diplomatic tracks and being able to "triangulate between the allies", Mount said.
The original decision to abandon the historic summit blindsided South Korea which had been brokering a remarkable detente between Washington and Pyongyang after months of Kim and Trump trading insults and threats of war.
North Korea had indicated it wasn't happy with the way things were progressing when it called of high level talks with South Korea over joint US-South Korean military drills. "Only time [and talent] will tell!" the president tweeted. The story also insinuated that Kim could be manipulating Donald.
The summit had been expected to take place in Singapore on 12 June.
Analysts say Kim's diplomatic outreach in recent months after a flurry of nuclear and missile tests in 2017 indicates he is eager for sanctions relief to build his economy and the global legitimacy the summit with Trump would provide.