Then aides to each were to come in for more discussions and a working lunch.
Trump has said the meeting could go one, two, or even three days.
Reports describing hard pre-summit negotiations at the working level about the contents of a joint communique-even on the most basic question, like what the term "denuclearization" actually means-offer evidence that the result of the Trump-Kim meeting will be modest.
He said Secretary of State Mike Pompeo and Defence Secretary James Mattis should oppose the one-on-one meeting "in writing".
"It's related to the North Korea-U.S. summit, so if anyone tries to go near the Yalu River, he or she will be accused of anti-regime crimes", the source said, referring to the waterway on the border between the North and China that is a popular route with defectors mainly when it freezes during the winter.
"Trump spoke yesterday with Prime Minister Shinzo Abe of Japan. We will be fine!"
At the conclusion of the summit, President Trump will participate in a media availability before departing for the U.S. at around 8 pm Singapore time.
Despite his young age - officially unknown but believed to be 34 -, Kim has stamped his authority on the North Korean leadership, reportedly undertaking a purge of top officials, including his powerful uncle Jang Song Thaek in December 2013 and the armed forces chief Hyon Yong Chol in April 2017. The president planned to stop in Guam and Hawaii on the way back to Washington.
Trump took keen interest in the pageantry of the day, insisting the pictures beamed around the world reflect a commanding leader making a decisive, world-altering move. "I could see Trump giving up a lot for very little in return".
Tweeting from his hotel room in the pre-dawn hours on Tuesday, Trump sought to heighten the suspense. The optimistic summit was a remarkable change in dynamics from less than a year ago, when Trump was threatening "fire and fury" against Kim, who in turn scorned the American president as a "mentally deranged USA dotard".
The United States are demanding that North Korea denuclearise in exchange for security guarantees. When Kim meets Trump, this will no doubt be on his mind. The South Korean press chased a North Korean cameraman through the streets when he left his hotel.
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The Justice Department had repeatedly suggested the judge consider forcing AT&T to divest some of Time Warner's cable channels. The Trump administration opposed the merger while the justice department has sued to block it.
"North Korea has previously confirmed to us its willingness to denuclearise and we are eager to see if those words prove honest", he said.
For Trump, achieving a momentous foreign policy success would cement his place in history, and give his presidency a boost just months ahead of congressional mid-term elections.
As he was trying to build a bridge with Kim, he was smashing longtime alliances with Western allies with his abrasive performance at the G-7.
He continued: "But in the end, that doesn't matter".
The 45-minute talk between Trump and Kim was held privately.
The pledge on denuclearization did not include the USA demands for a "complete, verifiable, irreversible dismantlement" of North Korea's nuclear weapons programs, she pointed out.
Mattis added that discussions about US troop numbers on the peninsula would be "premature" ahead of the outcome of negotiations between the two leaders Tuesday.
"I think of myself as American", he said. "We are prepared to take what will be security assurances that are different, unique, than America's been willing to provide previously".
There's excitement surrounding the summit but also skepticism that the North will relinquish a nuclear weapons program it spent decades building despite crushing sanctions.
"We're not going to go in and sign something on June 12th and we never were", Trump told White House reporters on June 1.
"I don't believe" troop levels are on the agenda, Mattis told reporters at the Pentagon on Monday.