In an interview that will air on "Hannity" Tuesday at 9:00pm ET, Trump was asked about some of the controversial remarks he had previously made, including calling Kim "Little Rocket Man" and saying that North Korea's continued threats against the USA would be "met with fire and fury like the world has never seen".
Mr Trump and Mr Kim also promised to work towards building "a lasting and stable peace" on the Korean peninsula, and to repatriate remains of prisoners of war and those missing in action during the Korean War.
His comments echoed those of US Secretary of State Mike Pompeo who said on Monday talks between the countries were "moving quite rapidly and we anticipate they will come to their logical conclusion even more quickly than we had anticipated".
It was signed by Trump and Kim shortly after they met privately and with their respective administrations, with the paperwork stipulating that Kim has 'reaffirmed his firm and unwavering commitment to complete denuclearization of the Korean Peninsula'.
In many ways, Trump's whirlwind negotiations with the North Koreans defy comparisons.
The two leaders concluded their Tuesday meeting by signing an agreement that acknowledges progress in their talks.
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Complete denuclearisation must mean the irreversible dismantlement of Pyongyang's nuclear weapons program.
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But despite the rows over trade and Russia , Mr Trump rated his relationship with other G7 leaders as 10 out of 10. As we reported president trump did depart the g7 early leaving behind festering tensions with our closest allies.
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The Korean Central News Agency and the Korean Central Television reported Monday that Kim departed from Pyongyang on Sunday morning via a Chinese chartered airplane to hold a summit with his USA counterpart in Singapore. If things turn out well, Trump will be encouraged to pursue similar tough terms with Iran.
Warmbier, 22, was a student at the University of Virginia when he was arrested in North Korea in January 2016 for allegedly stealing a propaganda poster. "I believe that he's going back and will start it virtually immediately", Trump said in excerpts of the interview broadcast on Fox after the U.S.
But it turns out, the issue was discussed and will be again in the future.
"Without support from China, it's really hard for Washington and Pyongyang to press ahead on denuclearisation and the peace process", Cheng said.
"We're not there yet", he said.
"It is a great victory achieved by both the United States and the two Koreas, and a huge step forward for people across the world who long for peace", Moon said.
Intelligence reports suggested Warmbier had been beaten so badly while in custody that there were fears he had died, a USA intelligence official told The New York Times.