"Great progress being made", the president said in a tweet that included both a copy of Kim's letter, dated 6 July, and its translation.
"I firmly believe that the strong will, honest efforts and unique approach of myself and Your Excellency Mr President aimed at opening up a new future between the DPRK and the U.S. will surely come to fruition", Mr Kim wrote, referring to North Korea by its acronym.
"I firmly believe that the strong will, honest efforts and unique approach of myself and Your Excellency Mr. President aimed at opening up a new future between the DPRK and the USA will surely come to fruition", the letter said.
Trump has touted the return of remains as one of the successes of his historic June 12 summit with Kim in Singapore. The North Korean leader also agreed to repatriate the remains of USA troops who died during the Korean War six decades ago.
Whether the meeting - the first time leaders of the United States and North Korea have met - has paved the way for peace (and delivers Trump a Nobel) is hotly disputed.
North Korea has long trumpeted a denuclearisation goal, but one that it sees as a lengthy process of undefined multilateral disarmament on the entire Korean peninsula, rather than a unilateral dismantlement of its nuclear arsenal. The US did not say which countries were illicitly providing Pyongyang with the products.
Mr Kim had agreed during the summit with President Trump to recover "POW/MIA remains, including the immediate repatriation of those already identified". Although Pompeo called the meetings "productive", North Korea said the lack of emphasis on security guarantees was "regretful".
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And US Secretary of State Mike Pompeo was condemned by Kim's right-hand man Kim Jong chol for his "gangster"-like attitude during a summit in Pyongyang last weekend".
"North Korea apparently wants a USA general to appear at the table to quickly finalize the repatriation issue", the source said.
North Korean officials denounced the meeting at its conclusion, accusing the USA of having a "gangster-like" attitude in negotiations.
North Korean and USA teams, as well as United Nations officials, had been due to discuss the details of repatriation.
The Defense Department estimates there are up to 5,300 sets of USA service member remains still somewhere in North Korea.
But North Korea's delegation failed show up.
A USA official waited at the border truce village of Panmunjom but no North Korean official showed up, South Korea's Yonhap news agency said, citing a Seoul diplomatic source. -North Korea relations worsened over North Korea's nuclear weapons program.