South Korea is willing to build an economic community with its neighbour once the effort to root out Pyongyang's nuclear ambitions is completed, Moon said. "We will be ready".
Interested in North Korea?
Trump himself has remained upbeat about the outcome of the first summit between the leaders of the US and North Korea.
What was in Kim's letter?
U.S. Defense POW/MIA Accounting Agency (DPAA) director Kelly McKeague stated in an interview with Reuters that the agency is "hopeful these discussions on July 12 will lead to further discussions and negotiations directly with the North Koreans by which we can actually get down to the detailed planning".
Mr Kim says that "the epochal progress" in promoting bilateral ties "will bring our next meeting forward".
Pompeo characterized the talks as progressive, asserting that North Korea did not "push back" on denuclearization. He did not elaborate.
In the letter Kim describes his June 12 summit with Trump in Singapore, and the resulting joint statement agreed by both sides, as the "start of a meaningful journey".
In return, Washington agreed to halt U.S. military exercises in South Korea. "We had anticipated the USA side would come with a constructive idea, thinking we would take something in return", a foreign ministry official said in a statement.
"We call upon (North Korea) to fully implement its worldwide obligations; to eliminate its nuclear, chemical, and biological warfare capabilities and ballistic missiles, and abandon all related 'programs", the statement said.
The note from Kim comes after Mr Trump reportedly sent a special gift over for Kim with Mr Pompeo - a copy of Elton John's "Rocket Man".
He left with only an agreement to set up working groups to discuss aspects of the deal, including the planned meeting on returning the remains. Representatives of the two countries had been poised to meet in Panmunjom, which is regarded as the truce village on the inter-Korean border.
Speaking hours earlier at a press conference in Brussels, Belgium, Trump pointed to North Korea's suspension of nuclear weapons and ballistic missile testing in the past eight months as a sign of progress in efforts to denuclearize the regime. Pyongyang has asked the UN Command Armistice Commission to upgrade the talks to a higher level to include a USA general, South Korea's foreign ministry said.
Pompeo's reception in Pyongyang last week was chillier than during his previous two trips to the North Korean capital and he didn't meet with Kim Jong Un.
There are about 7,700 U.S. soldiers classified as missing in action from the war.
China is North Korea's traditional ally and main trading partner, and is key to maintaining sanctions on Pyongyang.
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