Trump talked to reporters after escorting Kim out of the Oval Office and posing for photos, along with Pompeo, before Kim's departure. "It went very well", President Trump told reporters on the White House lawn.
"I want it to be very meaningful", he said.
In the latest sign of hostility cooling down but hopes kept in check, Trump said he's unilaterally put a hold on new sanctions against the North "until the talks break down".
South Korea says building trust with North Korea is crucial amid a USA -led diplomatic push to persuade the North to give up its nuclear weapons.
Trump referred to the letter from Pyongyang as "interesting".
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"Come to Russia, we'll be very happy to see you", Lavrov told Kim in remarks released by the Russian foreign ministry.
Plans for the high-stakes sit-down in Singapore had been cast into doubt. Responding to a question, Trump said he had never cancelled the meeting. "We welcome the summits that already took place between Pyongyang and Seoul as well as planned meetings between North Korean and USA leadership".
"I look forward to the day I can take the sanctions off North Korea", Trump said.
Since taking power in 2011, Kim has not publicly met with another head of state in North Korea.
Trump has refused to publicly acknowledge whether he's spoken directly with Kim Jong Un ahead of the talks.
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The government says the changes would weaken Britain's negotiating position and is seeking to reverse them in the Commons. But Solicitor General Robert Buckland said while talks may "yield fruit" - he could not guarantee a change in policy.
"We're meeting with the chairman on June 12, and I think it's probably going to be a very successful - ultimately, a successful process".
The US secretary of state, who spoke with Mr Trump on Wednesday night and with National Security Adviser John Bolton early on Thursday, was accompanied by Andrew Kim, the head of a Central Intelligence Agency unit assigned to work on North Korea, and Mark Lambert, the head of the State Department's Korea desk.
If held, the Trump-Kim meeting would be the first summit between the two countries that have always been at odds over the North's pursuit of nuclear weapons and ballistic missiles capable of striking the U.S.
Trump said countries in the region - Japan and South Korea - were also involved in this.
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Minnesota Public Radio branded the raccoon with the hashtag #mprraccoon , which became the top trend on Twitter. The woodland creature also had its own Twitter account , with one tweet saying, "I made a big mistake".
U.S. officials with knowledge of the letter's contents said it did not make any noteworthy threats or concessions, and was seen as a positive gesture. He also expressed Moscow's support for an agreement Kim reached with South Korean President Moon Jae-in at a summit last month that focused on measures to ease hostilities and increase exchanges between the two Koreas.