Trump, speaking after a landmark summit with Kim on June 12 in Singapore, said that both leaders were "prepared to start a new history and write a new chapter between our nations".
Before signing what Trump described as a "comprehensive letter", Kim said the two leaders had a historic meeting "and chose to leave the past behind".
Trump said he had formed a "very special bond" with Kim and that relationship with North Korea would be very different. At the time, the Koreans faced criticism for essentially kicking the issue of North Korea's nuclear arsenal down the road to Tuesday's Trump-Kim summit.
It did not include an agreement to take steps toward ending the technical state of warfare between the USA and North Korea.
President Trump says the nuclear threat from North Korea has entered a "new phase", a day after Pyongyang test-launched four ballistic missiles towards Japan. Trump called the drills "provocative and inappropriate", adding that he ultimately wants to "bring our soldiers home" - a reference to some 25,000 USA troops stationed in South Korea.
But Trump, who had once threatened to walk away from his negotiations with Kim if they appeared to be fruitless, allowed the dictator to leave Singapore with few hard commitments.
"Yeah, they are going to stay. Getting a good picture everyone, so we all look nice and handsome and thin.perfect".
The two leaders sign the summit declaration, watched by the North Korean leader's sister and US Secretary of State Mike Pompeo.
The pair promised in the document to "build a lasting and stable peace regime" on the Korean Peninsula and to repatriate remains of prisoners of war and those missing in action during the Korean War.
Fed raises interest rates, says two more hikes likely in 2018
The Fed's new projection for the pace of rate hikes shows four this year , three in 2019 and one in 2020. They see another three rate increases next year , a pace unchanged from their projections in March.
The document makes no mention of any move to end the technical state of warfare that persists between the United States and North Korea some 65 years after the conflict, which ended with a truce but no peace treaty. He left the summit site before Trump's news conference and didn't address reporters on his own. "And it will. It will soon end".
But, today they were seen shaking hands in a foreign land discussing two powerful countries.
Trump, beaming over what seemed nearly a bromance with one of the world's most brutal dictators, said he would invite Kim to the White House.
The signing ceremony followed a series of meetings at a luxury hotel on Singapore's Sentosa island, including a 45-minute face-to-face meeting between Trump and Kim and broader talks between their delegations. He added Otto Warmbier, an American once detained in North Korea, "did not die in vain" because his death brought about the nuclear talks. Trump told reporters after the summit that Kim had accepted an invitation to visit the White House, and that he would be willing to make a trip to Pyongyang at the "appropriate time". Trump had originally agreed to the summit, then called it off before reviving it again.
"Many people in the world will think of this as a form of fantasy. from a science fiction movie", Kim said via the translator after the two leaders first met.
Trump said he expected the denuclearization process to start "very, very quickly".
He and Kim "got along very well", Trump told VOA. "Glad I fired him on Apprentice!"
"Top of the line", the President said, arguing the summit went better than his critics had expected.
The two leaders pose for photographers from a balcony on the Capella Hotel in Singapore.
Bitcoin's soaring value down to manipulation, scientists claim
Bitcoin isn't all it has been cracked up to be, according to a new report by a University of Texas professor. Other digital currencies like Ethereum and Litecoin soared at around the same time by as much as 64 percent.
The first episode of this close encounter with Kim had the whole world binge-watching, from the first handshake to the dictator's tour of the presidential limousine, known as "The Beast", courtesy of Trump.
Late Tuesday, Trump took to Twitter to say there was "no limit" to what North Korea could achieve "when it gives up its nuclear weapons and embraces commerce" and engagement with the world. Complete, verifiable, irreversible denuclearization, he told reporters, "is the only outcome that the United States will accept".
Trump announced that he will be freezing USA military "war games" with its ally South Korea while negotiations between the two countries continue.
December 22, 2017: The UN Security Council unanimously adopts Resolution 2397, imposing additional sanctions on North Korea, including cutting refined petroleum imports by almost 90 per cent.
"Sanctions are not an end", Chinese Foreign Ministry spokesman Geng Shuang said at a daily briefing in Beijing.
But Trump, in his interview with VOA, reiterated his statement from the news conference that sanctions won't be lifted before results are seen.
"We're going to be announcing that in a couple of minutes", Trump said.
November 20, 2017: President Trump officially designates North Korea as a State sponsor of terrorism.
They shook hands for several seconds, Trump reaching out to touch the North Korean leader on his right shoulder.
Spain sack national coach Julen Lopetegui before start of World Cup
Guillem Balague says Julen Lopetegui's appointment as Real Madrid manager came as a surprise, but insists fans will be happy. The Spanish Football Association have sacked Julen Lopetegui just a day after he was named the new Real Madrid manager.