Expectations for the summit were being tempered by South Korean president Moon Jae-in, who said: "The deep-rooted hostile relationship and the North Korean nuclear issue can not be resolved in one single action in a meeting between leaders".
"We were able to complete the preparations for the historic summit, and I would like to thank you for that", Kim, speaking through an interpreter, told Lee.
For more on the meeting in Singapore, we cross to Oliver Miocic in Seoul.
In March, Trump surprised the world by accepting an invitation from Kim to meet in person.
The meeting between Kim and Trump will mark a turnaround of relations between the two leaders after a long-running exchange of furious threats and insults. He released a photograph from inside the room of North Korean vice foreign minister Choi Sun-hee smiling across the table at the USA lead negotiator, the Philippines ambassador Sung Kim.
The Rodong Sinmun newspaper, the mouthpiece of the ruling Workers' Party, devoted its first two pages and 16 photos to Kim's trip, including images of him boarding an Air China Boeing 747 for the journey.
"Wide-ranging and profound views on the issue of establishing new DPRK-U.S. relations, the issue of building a permanent and durable peace-keeping mechanism on the Korean Peninsula, the issue of realizing the denuclearization of the Korean peninsula and other issues of mutual concern, as required by the changed era, will be exchanged at the DPRK-U.S. summit talks", a dispatch by the North's official Korean Central News Agency (KCNA) said.
The comment can be dismissed as casual optimism characteristic of the U.S. leader, but the watchers interpreted it as suggesting that the atmosphere in their last-minute negotiations on denuclearization may not be discouraging.
The United States and North Korea have been in a technical state of war for decades after the eruption of the 1950-1953 Korean War, which pitted U.S. -led U.N. troops backing Seoul against Pyongyang's forces that were aided by China.
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Trump said after a meeting with North Korean envoy Kim Yong-chul at the Whitehouse.
"He has that opportunity, and he won't have that opportunity again". And based on the tweets that Trump sent Monday morning, it doesn't seem like he left his baggage from the first summit at the door.
White House press secretary Sarah Huckabee Sanders said Trump spent his flight from Canada to Singapore "meeting with his staff, reading materials and preparing for his meetings in Singapore".
Trump's administration has insisted that CVID is a foremost requirement that Pyongyang must meet for lasting peace on the Korean peninsula.
Earlier Trump had said the meeting was a "one-time shot" for Kim.
Despite the initial high stakes of a meeting meant to rid North Korea of its nuclear weapons, the talks have been portrayed by Trump in recent days more as a get-to-know meeting.
Pyongyang has said it is willing to deal away its entire nuclear arsenal if the United States provides it with reliable security assurances and other benefits.
The summit has also raised hopes of progress towards a peace treaty to formally end the Korean War, the last festering legacy of the Cold War, after hostilities only stopped with an armistice. But there are major doubts, given how hard it has been for Kim to build his program and given that the weapons are seen as the major guarantee to his holding onto unchecked power. China and South Korea would have to sign off on any legal treaty.