The joint declaration states that the USA has committed to providing "security guarantees" to Pyongyang.
For his part, South Korean President Moon Jae-in said on Monday that he hopes Trump and Kim "generously" consider each other's demands and meet the people's wish for a new era for the Korean Peninsula.
The United States and DPRK will join their efforts to build a lasting and stable peace regime on the Korean Peninsula.
The July 12th meet would be followed by talks between the US Secretary of State Mike Pompeo and senior North Korean officials "at the earliest possible date" to implement the outcomes of the Singapore summit. Very smart, very good negotiator.
With cameras of the world's press trained on them, Trump and Kim displayed an initial atmosphere of bonhomie.
Moment Kim Jong-un and Donald Trump share historic handshake - video 2. Assuming that nuclear war doesn't break out.
Trump said Tuesday after meeting Kim that he's feeling "really great".
They said the summit aimed to work towards the "promotion of peace, prosperty and security of the Korean Peninsula and of the world".
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He is now entered to play the Queen's Club tournament next week. "Of course it's a drastic change from clay to grass". Nadal's victory on Sunday earned him a 17th slam, putting him just three behind Federer's record haul of 20.
Trump and Kim have a long day ahead of them.
Trump has a point, and good eye for prime real estate.
Then they dined on beef short rib confit along with sweet and sour crispy pork.
Mr Trump added: "I learned he's a very talented man".
After shaking hands again with Kim in front of the cameras, Trump gave a thumbs-up before starting the first meeting between an incumbent us president and a DPRK leader.
Mr Trump said he had formed a "very special bond" with Kim and that relationship with North Korea would be very different. Kim has been accused of horrific rights abuses against his people.
Before the history-making meeting Trump taunted his critics on Twitter.
He said the Canadian leader must not have realised that Trump had televisions on Air Force One, allowing him to monitor Trudeau's news conference at the end of the G-7 summit.
"We will be doing something on it", Trump said.
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However, they face a revolt by pro-EU Tory MPs determined to retain as numerous changes as possible in the legislation. It is thought a second ministerial resignation could follow.
Employees at a foreign exchange trading company in Tokyo watch the meeting.
He also revealed that his economic adviser, Larry Kudlow, who did not among the US delegation in Singapore, had suffered a heart attack was being treated at Walter Reed Medical Center in Maryland.
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".
Many across Asia and around the world have welcomed a flurry of diplomacy in recent months between the two adversaries, following a year of mounting tension and threats.
Lifting sanctions could be a win for North Korea - possibly enough to show some success from this first meeting in Singapore, Gregg added.
It was not immediately clear what specific progress, if any, had been made in preliminary discussions between USA and North Korean officials.
"The discussions between the United States and North Korea are ongoing and have moved more quickly than expected", the White House said in a statement. "I got a lot of people together, and Kim Jong-un told me this was the first time anybody's ever kept their word to me and my country", Rodman recalled.
Mr Trump and Mr Lee had a private, one-on-one meeting that lasted about half an hour.
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Listen to Newsbeat live at 12:45 and 17:45 every weekday on BBC Radio 1 and 1Xtra - if you miss us you can listen back here. While many fans tweeted their congratulations, comedian Kathy Griffin was a little more cautious in her response...