"Great progress being made", Trump tweeted on Thursday along with the letter, calling it a "very nice note from Chairman Kim".
The North Korean leader affirmed his firm belief that "the strong will, honest efforts and unique approach" the two have "aimed at opening up a new future between the DPRK and the USA will surely come to fruition".
This latest incident comes shortly after Pompeo's visit to Pyongyang resulted in North Korea accusing Pompeo of making a "unilateral and gangster-like demand for nuclearization".
Kim committed to "complete denuclearisation" when he and Trump met in Singapore last month.
Pause in op to rescue Thai boys from cave
Authorities in northern Chiang Rai province began the unsafe mission to bring out the 12 boys and their coach earlier on Sunday. After emerging from the cave, they were taken in an ambulance to be evacuated from the cave site via helicopter.
Trump has expressed an eagerness to tout the recovery of the war dead, telling Fox News that Kim was "giving us back the remains of probably 7,500 soldiers".
North Korean officials did not show up to a planned meeting with US officials at the demilitarized zone Thursday, where they were expected to continue discussions on repatriating the remains of Americans killed during the Korean War, a senior USA official tells CNN.
He was referring the summit statement signed by the two leaders - the first-ever US-North Korean summit - in which Kim agreed in broad terms to "work toward denuclearization" of the Korean Peninsula. "We will be ready". Trump tweeted on Thursday alongside a copy of the letter, dated July 6.
Attached were two images of the letter - one in Korean and the other an English translation.
A planned meeting Thursday between North Korean and US officials in the Demilitarized Zone between the two Koreas to discuss the return of the remains was postponed.
A day-by-day look at the Thailand cave ordeal
Two of the eight rescued so far have shown signs of a potential lung infection, but there is a hope they could honour the invite. All they wanted to talk about was whether there were any updates about the team. "Just returned from Cave 3", Musk said.
One problem - no North Korean officials showed up, The New York Times reported. The U.S. also risks giving the North Koreans leverage to continue diplomacy and drag out disarmament talks. Reports say North Korea is now asking for higher-level talks for Sunday.
While the details are unclear, a setback on talks over the remains wouldn't bode well for broader negotiations about North Korea's nuclear program.
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State Department spokeswoman Heather Nauert said North Korea called at midday Thursday and offered to meet on Sunday instead.
Trump himself has remained upbeat about the outcome of the first summit between the leaders of the USA and North Korea.
Serena rolls past Goerges to return to Wimbledon final
History was made at Wimbledon this year when all the top-10 seeds were knocked out before the quarterfinals for the first time. Williams will next face German world No 10 Angelique Kerber on Saturday in a repeat of the 2016 showpiece won by Williams.
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.