The pledge to return the remains of USA soldiers was made during a landmark summit between Trump and Kim in June in Singapore, where North Korea committed to work toward the complete denuclearization of the Korean peninsula.
Earlier Thursday, Trump thanked Kim on Twitter.
'The ongoing correspondence between the two leaders is aimed at following up on their meeting in Singapore and advancing the commitments made in the US-DPRK join statement, ' she said in a statement Thursday.
The White House did not provide details on the specific content of the letter, which was received Wednesday.
Vice President Mike Pence and USA military leaders in Hawaii on Wednesday received the remains of missing US soldiers from the Korean War.
Pence also praised Trump for having the nation's fallen war heroes "on his heart" during his summit with Kim, which was focused on denuclearization of the Korea Peninsula.
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Pence, in his address at Joint Base Pearl Harbor-Hickam in Hawaii, said he trusted that Americans killed in the war were among the flag-draped cases flown to Hawaii on Wednesday.
The pledge to transfer war remains was seen as a goodwill gesture by Kim at the Singapore summit, and was the most concrete agreement reached by the two sides so far.
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The remains are being sent to Defense POW/MIA Accounting Agency for forensic analysis in order to try to determine their identities.
Experts say that the painstaking identification process could take anywhere between days to decades to complete. He said during a Tuesday rally in Tampa, Florida, that the USA was "doing well" with North Korea and noted the return of detained Americans and Pyongyang's ceasing of nuclear testing and missile tests.
Air Force General John Hyten, who as commander of the U.S. Strategic Command oversees America's nuclear forces, said North Korea had clearly demonstrated its commitment to halt flight-testing missiles since none had launched since November. It was the first repatriation of remains since 2005.