At the Singapore summit on 12 June, Mr Trump and Mr Kim signed a document that included a pledge from North Korea to rid the Korean peninsula of nuclear weapons.
I deeply appreciate the energetic and extraordinary efforts made by Your Excellency Mr. President for the improvement of relations between the two countries and the faithful implementation of the joint statement.
Kim refers to Trump as "your excellency" five times, praises the "improvement of relations" between the USA and North Korea, and expresses hope for a new future.
The statement was carried by North Korean state-run press right after U.S. State Secretary Mike Pompeo finished his two-day visit to the communist country.
Kim made that commitment when he met Trump in Singapore last month.
An estimated 30,000 USA soldiers died in the Korean war.
And even as Trump released Kim's note, the USA was asking the U.N. Security Council to punish North Korea after finding that Pyongyang had violated restrictions on refined oil imports. Pyongyang has asked the UN Command Armistice Commission to upgrade the talks to a higher level to include a US general, South Korea's foreign ministry said.
Pompeo called the talks "productive", although North Korea said it was disappointed with the emphasis on denuclearization with a lack of security guarantees.
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The UNC told the USA defense ministry about North Korea's proposal and is waiting for Washington's answer, the source added.
Trump, on Thursday afternoon, tweeted out a letter he received from Kim Jong Un, along with a translated version.
Earlier Thursday the North Koreans postponed a meeting with USA representatives at the inter-Korean border meant to discuss the repatriation of remains of American troops killed in the Korean War.
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The Defense Department estimates there are up to 5,300 sets of US service member remains still somewhere in North Korea.
Trump has sought to cast his summit with Kim as a major victory, although reports have suggested North Korea is continuing with its nuclear programme and promises of "denuclearisation" remain vague.
"North Korea apparently wants a US general to appear at the table to quickly finalize the repatriation issue", the source said.
"The U.S. side had had discussions with the North with an aim to hold the talks as early as July 12, but it appeared that the North was not ready for the talks", the source told Yonhap News Agency on condition of anonymity.