There were mo immediate details on the contents of the document.
Should they succeed in making a diplomatic breakthrough, it could bring lasting change to the security landscape of Northeast Asia, like the visit of former US President Richard Nixon to China in 1972 led to the transformation of China.
"We had a historic meeting and made a decision to leave the past behind and we are about to sign the historic document", Kim said through a translator.
"People are going to be very impressed and people are going to be very happy and we are going to take care of a very risky problem for the world", Trump said.
Mr Trump's delegation at the luncheon included US National Security Adviser John Bolton, Secretary of State Mike Pompeo, White House Chief of Staff John Kelly and Press Secretary Sarah Huckabee Sanders.
North Korea's supreme leader seeks a formal end to that war, which has frozen the Korean peninsula in a state of conflict throughout the modern era and has kept almost 29,000 U.S. troops indefinitely stationed in the south, as well as an injection of foreign direct investment.
"It's interesting because we believe Kim speaks English - he was educated in Switzerland and speaks German, but Trump seems to be the only one talking during their bit of chit chat", Professor Hajek told Yahoo7.
It is the first time a sitting USA president has ever met a North Korean leader.
Meanwhile, Law and Home Affairs Minister Shanmugam is confident that security is in place for the Trump-Kim summit.
Kim, dressed in a dark suit, and Trump, sporting a red tie, talked in a corner of the hotel corridor for a few minutes before they walked along the passage into the meeting room and sat down in front of journalists.
Speaking through his translator, Mr Kim said it was "not easy" to get here.
"I'm very optimistic that we will have a successful outcome from our meetings tomorrow", he said.
And as they emerged from the meal for a brief stroll together, Trump appeared to delight in showing his North Korean counterpart the interior of "The Beast", the famed USA presidential limousine known for its high-tech fortifications.
The combatants of the 1950-53 Korean War are technically still at war, as the conflict, in which millions of people died, was concluded only with a truce.
Mr Trump added: "I learned he's a very talented man".
He later told Mr. Trump: "Many people in the world will think of this as a ... form of fantasy ... from a science fiction movie".
"Very good, very good".
Kim also sounded positive about the prospects of peace. "I came here after overcoming all obstacles".
Kim referred to the decades of hostility that preceded the landmark meeting.
The dollar jumped to a 3-week top on Tuesday and Asian shares rose on the news.