He said he also hoped to “bring home” the 32,000 US troops stationed in South Korea at some point in the future, but acknowledged such a move was not “part of the equation right now”.
In a press conference lasting more than an hour Mr Trump also said the North had begun dismantling a major missile engine testing site, and he praised Mr Kim as “very talented”.
Later, as Mr Trump's plane landed in Guam to refuel, Mr Trump told reporters that he trusted Mr Kim and that he believed the North Korean leader was sincere about his desire for denuclearisation.
“I can only tell you that from the time I’ve (dealt) with him, which is really starting 90 days ago," Mr Trump said. "I think he wants to get it done".
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At the end of the summitMr Trump and Mr Kim signed what Mr Trump claimed was a “comprehensive” document following the historic meeting aimed at the denuclearisation of the Korean peninsula. The president said he believed the process of denuclearisation would happen “very, very quickly”, adding that he had formed a “special bond” with Mr Kim.
Mr Kim who has been granted a measure of international legitimacy with the summit, said the two leaders had “decided to leave the past behind. The world will see a major change”.
In the first meeting of a sitting US president and a North Korean leader, the pair convened at a luxury resort on Singapore’s Sentosa Island, clasping hands as they stood on a red carpet in front of a backdrop of alternating US and North Korean flags. Mr Trump was first to arrive at the summit site, followed by Mr Kim, both readying for the 9am meeting that culminated dizzying weeks of negotiations over logistics and policy.
The pair started the momentous Singapore summit with an historic handshake for the world’s media before getting down to talks about North Korea’s nuclear programme.
Mr Trump and Mr Kim met one on one for about 40 minutes, joined only by translators. Then aides to each arrived for more discussions and a working lunch.
The US president said the meetings went “better than anybody could have expected” after the pair emerged from lunch and strolled together down a paved walkway before stopping and posing before the waiting news media.
Mr Trump said the meeting is “going great. We had a really fantastic meeting”. He added that there has been “a lot of progress. Really very positive”
It is believed that the signing will likely revolve around a promise to keep meeting.
The White House said discussions with North Korea have moved “more quickly than expected” and Mr Trump would leave Singapore on Tuesday night, after the summit. He had earlier been scheduled to leave on Wednesday. Mr Trump will visit military bases in Guam and Hawaii on his way back to Washington.
Teams of officials from both sides held working-level talks on Monday.
Senior officials travelling with Mr Trump included secretary of state Mike Pompeo, national security adviser John Bolton and White House chief of staff John Kelly. As Mr Trump was travelling to the Capella Hotel which was the site of the summit, he surprisingly tweeted about another senior official - economic adviser Larry Kudlow - with Mr Trump saying he had had a heart attack. The White House later said that Mr Kudlow was in a good condition in hospital having suffered a “very mild” heart attack.
Mr Kim’s delegation consisted of foreign minister Ri Yong Ho, defence minister No Kwang Chol and Kim Yong Chol, a close aide of Kim who has been instrumental in the diplomacy that culminated in Tuesday’s summit.
Kim Yo Jong, leader Kim’s younger sister, was also spotted in his delegation. She emerged as an influential figure in Pyongyang’s opaque leadership in February, when she led a North Korean delegation to the Winter Olympics in South Korea.
When Mr Trump initially agreed to meet with the North Korean leader, the US president spoke of his hope that their encounter could secure a major breakthrough and lead to the denuclearisation of the Korean peninsula.
If so, then the meeting would be the most important since Ronald Reagan met with Soviet leader Mikhail Gorbachev in Geneva in November 1985.
Mr Trump has since sought to play down expectations, saying that the meeting will be an important first step, but that securing a deal will likely take many more meetings.
Given that what the US wants to get out of the summit, a rapid denuclearisation of the Korean peninsula, may be different to what North Korea wants, there may be many such meetings. Many observers will be looking to see whether Mr Trump does extend an invitation to his counterpart to visit the White House.
Welcome to The Independent's live blog from Singapore on the historic summit between Kim Jong-un and Donald Trump.
The leaders arrived late on Sunday and both were received at the airport by Singapore’s foreign minister, Vivian Balakrishnan, before heading to their respective luxury hotels.
"Great to be in Singapore, excitement in the air!" Mr Trump tweeted on Monday morning. He will have lunch with the Singaporean prime minister at the Istana, the country's presidential palace.
Mr Kim has no official agenda for Monday, having already met prime minister Lee Hsien Loong on Sunday, hours after arriving on a plane loaned to North Korea by China.
Mr Trump and Mr Kim will meet in front of the world's media at around 9am local time on Tuesday. Any progress will have to be swift - the North Korean delegation is reportedly scheduled to leave Singapore as early as 2pm on Tuesday.
Teams of officials from both sides will hold working-level talks on Monday, with the aim of preparing a joint statement or agreement Mr Trump and Mr Kim can sign.
White House spokeswoman Sarah Huckabee Sanders said the US delegation would be led by Sung Kim, a veteran diplomat who recently held talks with North Korean officials.
But another US official, speaking to Reuters on condition of anonymity, said Monday's meeting appeared aimed at making 11th-hour progress ahead of the summit, since Sung Kim's earlier talks did little to narrow a gap between the two sides on the definition of denuclearisation or win agreement on tangible commitments from Pyongyang toward dismantling its nuclear arsenal.
Mr Trump arrived in the country's Paya Lebar Air Base aboard Air Force One, having flown directly from the turbulent G7 meeting in Canada, where he fought with his supposed allies and refused to sign a joint communique.
Those hoping Mr Trump’s meeting with the North Korean leader helps bring about peace to the region and wider world, will be praying the US president is more diplomatic with Mr Kim than he was with Canadian prime minister, Justin Trudeau, who he called “very dishonest and weak”.
When he landed at around 8.20pm on Sunday, Mr Trump was asked by a reporter how he felt about the upcoming meeting, for which he has reportedly prepared minimally. He replied: “Very good.” His motorcade, including the reinforced “Beast” Cadillac limousine, made its way to the Shangri-La hotel, where a sweating line of photographers and a small crowd of people were waiting to see him pass.
“It’s exciting, but I am also anxious,” said a woman who asked to be identified only by her first name, Kim. “Kim can change his mind at any time. Mr Trump also likes to get his own way.”
She added: “But if it works, it’s for the good of the whole world. It will make history.”
Christine McDougal, a Singaporean resident and US citizen, had arrived with a friend to cheer the president’s arrival. “This is a such an important moment,” said the 16-year-old student.
Mr Kim had arrived earlier in the afternoon, touching down at the nation’s main airport, Changi, in a plane loaned to him by China. While North Korea has proved it has been able to create a successful intercontinental ballistic missile, there are questions about the air worthiness of its aircraft for anything other than short journeys. Like Mr Trump, Mr Kim was received at the airport by Singapore’s foreign minister, Vivian Balakrishnan.
He then made his way to the St Regis, one of the nation’s most exclusive and expensive hotels. It is understood that Singapore has agreed to meet the costs of Mr Kim’s bill.
While the two hotels were surrounded by media on Sunday, it will be the Capella hotel on the island of Sentosa, half a mile off shore, where all attention will turn on Tuesday. It is there, in a building designed by British architect Norman Foster, that Mr Trump and Mr Kim are due to meet, starting at 9am.
Mr Trump, speaking in Canada on Saturday, said any agreement at the summit would be "spur of the moment," underscoring the uncertain outcome of what he called a "mission of peace".
He initially touted the potential for a grand bargain with North Korea to rid itself of a nuclear missile programme that has advanced rapidly to threaten the United States.
But he has since lowered expectations, backing away from an original demand for North Korea's swift denuclearisation.
He has said the talks would be more about starting a relationship with Kim for a negotiating process that would take more than one summit.
North Korea's official media reported for the first time this morning what the rest of the world knew when he arrived yesterday - that leader Kim Jong-un is in Singapore to meet President Donald Trump to discuss how to forge a new relationship.
The first word Mr Kim is in Singapore for Tuesday's summit came early Monday morning from the state-run Korean Central News Agency (KCNA). Though North Koreans have been left largely in the dark about the summit, the reports of his arrival came relatively quickly by North Korean standards.
A dispatch by KCNA says Mr Kim and Mr Trump will exchange "wide-ranging and profound views" on establishing a new relationship, the issue of building a "permanent and durable peace mechanism" and realising the denuclearisation of the Korean Peninsula.
The report notes the summit is being held "under the great attention and expectation of the whole world."
Donald Trump arrived at Singapore's government house for a meeting with the island city-state's leader shortly before before noon local time.
Prime Minister Lee Hsien Loong are meeting one on one before they're joined by aides for a larger working lunch.
In October, Mr Trump praised the US relationship with Singapore and called Mr Lee a "wonderful and loyal partner."
Waiters were spotted near an elaborate dining room carrying Diet Cokes. That's Trump's beverage of choice.
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