After the summit, Trump declared that the North is "no longer a nuclear threat".
The inter-Korean summit would come just before a gathering of world leaders at the United Nations in NY at the end of September, but Seoul said Thursday that it was unlikely Kim would attend.
Earlier Thursday, North Korean media said that Kim Jong Un had called for further efforts to "realize the denuclearization of the Korean Peninsula" during meetings with the envoys.
Kim said such a declaration would not imply a withdrawal of American forces or a weakening of the US-South Korean military alliance, Chung said. "There's been some rhetoric, let's see what happens", Trump said.
For a long time, the United States and North Korea were at an apparent impasse, with the U.S. asking for a comprehensive list of North Korea's nuclear facilities, and Pyongyang demanding a formal end to the Korean War.
Under discussion is whether denuclearisation or declaring an end to the 1950-53 Korean War should come first. The UN General Assembly in late September would be an ideal date for Seoul, but many analysts see that possibility as low, considering the complications of the process and how far apart the parties now are.
The primary sticking point at the moment is North Korea's desire to bring an official end to the Korean War.
U.S. Secretary of State Mike Pompeo visited Pyongyang in July, after which North Korea accused him of making "unilateral and gangster-like demands for denuclearization" while showing little interest in ending the war.
According to South Korean officials, North Korean dictator Kim Jong Un wants to complete the denuclearization of his country by the end of President Donald Trump's first term. In other words, the North may not move any step forward without a cause like a declaration of an end of the war.
The development followed a recent halt in negotiations between Washington and Pyongyang.
Kim reportedly also did not have anything negative to say about Trump, and expressed that he wanted his relationship with the U.S.to continue to improve.
Earlier this week, Moon sent special envoys to Pyongyang to help resolve the nuclear stalemate.
One Security Council member, China - which is also our biggest trading partner - would be most upset if New Zealand was helping the USA board its ships, or those of its friends, in North Asian waters.
Kim noted the dismantlement of his country's only known nuclear test site in Punggye-ri has left the country with no means to stage any more nuclear tests.
North Korean children perform at the May Day stadium during the "Arirang" mass games song-and-dance ensemble on the eve of the 60th anniversary of the Korean War armistice in Pyongyang, North Korea.
A convoy of buses carrying participants of an inter-Korean family reunion makes its way past a South Korean guard post through the Demilitarized Zone (DMZ) towards North Korea, in Goseong on August 20, 2018.