The letter is dated July 6, 2018.
He shared the four-paragraph letter on Twitter, in which Mr Kim describes their 12 June summit - and the resulting joint statement - as the "start of a meaningful journey".
Kim voices hope that "the invariable trust and confidence in Your Excellency Mr. President will be further strengthened in the future process of taking practical actions".
Trump's comments came just hours after North Korean officials failed to show up at a planned meeting Thursday with their USA counterparts to discuss the repatriation of the remains of American soldiers killed in the Korean war.
Kim also agreed to repatriate remains of United States troops who died during the Korean War six decades ago.
"We have to see the USA defense ministry's response, but the UNC reportedly gave a positive signal to North Korea, so there is a good chance of the talks taking place on Sunday", the source said.
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After his side was shown six yellow cards, Colombia coach Jose Pekerman made clear his unhappiness with US referee Mark Geiger. I apologise to the Colombian people, but the players are not to blame as we don't choose the referees.
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Other speculation is that North Korea was discomforted with the way it was portrayed by the US media _ as trying to make money from the war remains.
"We were ready", the official said.
The North, instead, reportedly proposed holding general-level military talks with the United Nations Command (UNC) to discuss returning the war remains to the U.S. It is unknown whether the North Korean side gave any prior notification to the U.S. about this.
USA officials and experts have said the shift in attitude was to be expected as the North Koreans are known for brinkmanship and trying to up the ante ahead of tough negotiations over their nuclear missile program. "The U.S.is fatally mistaken if it went to the extent of regarding that [North Korea] would be compelled to accept, out of its patience, demands reflecting its gangster-like mindset".
The Trump administration has insisted that progress is being made.
Though Mr. Trump claimed on June 13 that North Korea was no longer a nuclear threat, just a day after his meeting with Mr. Kim, experts are not sure if North Korea is following through on the deal.
Kim made that commitment when he met Trump in Singapore last month.
On Sunday, Pompeo seemingly shrugged off the bombastic statement and said the talks with North Korean officials were conducted in "good faith".
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