SINGAPORE - US President Donald Trump said North Korean leader Kim Jong Un gave his "unwavering commitment" to work towards denuclearisation during their historic summit in Singapore on Tuesday (June 12), and they are prepared to "write a new chapter" between their countries.

Taking questions at a press conference for over an hour, the American president was clearly in good spirits - often praising Mr Kim, Chairman of the State Affairs Commission, and making clear that he thought Mr Kim was keen to bring change to his country.

"Chairman Kim has before him an opportunity like no other to be remembered as the leader who ushered in a glorious new era of prosperity for his people," Mr Trump said. "He reaffirmed his unwavering commitment to the complete denuclearisation of the Korean peninsula."

We are "prepared to start a new history and write a new chapter between our nations", he added.

Mr Trump also dismissed suggestions that North Korea had come out better off from the meeting, having secured a suspension of joint US-South Korea military exercises in exchange for a denuclearisation pledge that did not come with concrete deliverables or targets.

"I'm here for one day. The process is now going to take place," he said, adding later that the two sides have already set up a follow-up meeting between officials next week.

He noted that complete denuclearisation takes time and sanctions would not be lifted until it happens. "The sanctions will come off when we are sure the nukes are no longer a factor," he said.

Mr Trump also said that Mr Kim agreed to destroy his country's "major missile engine testing site" even though it was not part of the agreement.

And in a sign of just how positive the American leader was, he refused to be drawn on the military options still on the table should Pyongyang renege on its promises, adding that he trusted Mr Kim.

The two leaders had signed a comprehensive document capping their historic talks earlier in the day at the luxury Capella Singapore hotel on Sentosa island.

Mr Trump described the talks as honest, direct and productive, saying it had achieved more than what he had hoped for.

Besides the nuclear issue, the two leaders also discussed the issue of the remains of prisoners of war. Mr Trump said Mr Kim agreed to hand over the remains.

The two leaders appeared to have developed a rapport over the course of a morning filled with remarkable political theatre.

The day had started with a historic handshake. At 9.04am, Mr Trump and Mr Kim strode into the courtyard from separate sides, stood face to face for the first time and shared a 12-second handshake against a backdrop of American and North Korean flags.

They then sat down for a 41-minute one-on-one meeting, after which they appeared before the media, with Mr Trump saying the discussion was "very very good".

This was followed by an extended bilateral meeting which included delegations from both sides and a working lunch.

Speaking to the media after the signing ceremony, Mr Trump said: "We had a terrific day and we learnt a lot about each other and our countries."

Calling Mr Kim "a very worthy, very smart negotiator", he said: "He's a very talented man, I also learnt that he loves his country very much."

He added: "We'll meet again, we'll meet many times".