The question is whether it will be enough for President Donald Trump to pick up where Moon has left off. Trump told reporters Wednesday that the outcome of the summit was "very good news" and that "we're making tremendous progress" with North Korea.
Trump tweeted late Tuesday: 'Kim Jong Un has agreed to allow Nuclear inspections, subject to final negotiations, and to permanently dismantle a test site and launch pad in the presence of global experts.
Mr Pompeo said earlier he was ready to restart negotiations, stalled since the first Trump-Kim summit in June.
The hope is that it will kickstart talks between the U.S. and North Korea which have stalled.
"Invoking the end of Trump's term in January 2021 is another way of saying to the North that American patience is not unlimited and that Kim Jong Un won't be able to sidestep denuclearisation indefinitely", Russel added.
Members of Trump's administration are concerned that such a declaration could be used to undermine the justification for the presence of US forces in South Korea.
Speaking at a joint news conference in Pyongyang, the two Korean leaders agreed to turn the Korean Peninsula into a "land of peace without nuclear weapons and nuclear threats".
The two countries also agreed to engage "in constant communication and close consultations" to review the implementation of the previously signed Panmunjom Declaration "and prevent accidental military clashes by promptly activating the Inter-Korean Joint Military Committee".
Moon and Kim have sought to strengthen the North-South relationship at their Pyongyang summit, agreeing that the North Korean leader will visit Seoul "at an early date".
Plans were laid out for a reciprocal visit by Mr Kim to Seoul - which would make him the first North Korean leader to do so.
"We understand the North has modified the performance in consideration of the position of our delegation", he said, adding Pyongyang may change the title of the performance.
To hurry the denuclearization process, Kim said he wants Secretary of State Mike Pompeo to visit Pyongyang and to set up another summit with Trump "as soon as possible", according to a summary by the Pyongyang Press Corps.
South Korean President Moon Jae-in puts water from the crater lake into a bottle as his wife Kim Jung-sook watches.
Asked about the specific corresponding United States measures demanded by Kim for denuclearisation, Moon said the matter should be discussed between North Korea and the US.
North Korea and South Korea's leaders took to the road for the final day of their summit Thursday - heading to a lovely volcano considered sacred in the North - following a day of wide-ranging agreements they trumpeted as a major step toward peace on the Korean Peninsula.
"North Korea expressed their willingness to speed up their denuclearization actions if [the United States] takes actions to form a new U.S".
Moon and Kim took a cable vehicle together to Heaven Lake, a caldera at the top of the mountain, and walked around the area with their wives and officials from both sides.
Moon said that during his time with Kim, the North Korean leader discussed specific steps he was willing to take toward destroying his nuclear arsenal for the first time, and reiterated his commitment to denuclearization.
"I propose to take the next big leap to completely resolve the 70 years of hostility and be together as one again", Moon said.