The president also adds that without Otto Warmbier "this wouldn't have happened". Just as a reminder, Otto Warmbier was the student arrested in North Korea but who came home in a coma - his parents allege torture. Trump says the meeting was a vindication for his parents somehow.

Trump says the US is not reducing military capabilities as part of the assurances it will provide North Korea. But he says the US will be stopping its military exercises with South Korea - these routinely infuriate North Korea.

In response to a question Mr Trump says that Mr Kim is "very talented". He came to power at a young age and ran a country, and "ran it tough".

"We can replace the horrors of battle with the blessings of peace," Trump continues.

"There's no limit to what North Korea can achieve when it gives up its nuclear weapons," he says.

He says the rest of the world "really wants to engage with North Korea".

Kim could be "remembered as the leader who ushers in a new era of prosperity for his people", he says.

Trump says Kim is "already destroying a major missile site".

This detail is not in the joint declaration because the two leaders agreed it after the document was created, he says.

"Today’s the beginning of an arduous process – our eyes are wide open," he says.

Image source, Reuters

Trump says "real change is indeed possible".

"My meeting with Chairman Kim was honest, direct and productive," he says.

Trump has begun his news conference.

"We've had a tremendous 24 hours, a tremendous three months actually," Mr Trump says.

"It has the potential to be an incredible place," he says of North Korea.

Laura Bicker
BBC News Koreas Correspondent

The agreement that the two sides have signed is vague and lacking in detail. It only commits the United States and North Korea to the denuclearisation of the Korean Peninsula.

There is no mention that the process should be verifiable or irreversible - which is what the US has been pushing for. There is no timetable to disarm North Korea nor is there any invitation for weapons inspectors to be allowed in.

The two sides have promised to hold follow up talks and work on efforts to build a lasting peace regime. This could lead to a treaty which could end the near 70-year war on the Korean Peninsula - however it looks like a long term process.

But this is at least a start. Something to build on. The enduring image of this day will be of a usually reclusive North Korean leader shaking the hand of the US president. A powerful signal that this relationship between two decades old enemies has changed.

Malaysia's Prime Minister Mahathir Mohamad says his country will reopen its embassy in Pyongyang. The two countries were embroiled in a diplomatic row after the killing of Kim Jong-nam, Kim Jong-un's half brother, in 2017. There is widespread suspicion Pyongyang was responsible - something North Korea denies.

You will be aware by now that many analysts are pouring cold water on President Trump's claims that this agreement is "comprehensive".

They say North Korea has signed much more substantial declarations after past negotiations.

So what has the Kim family agreed to in the past - only to later renege?

• In 1992, the Koreas agreed to not test or produce nuclear weapons
• In 1994, North Korea said it would freeze its plutonium weapons programme in return for aid
• In 2005, it said it would abandon "all nuclear weapons and existing nuclear programmes"

In case you needed any further confirmation that Trump is happy with how the summit's gone - he's just tweeted a video of his meeting with Kim Jong-un - complete with uplifting music.

More analysts have been weighing in and their invective has grown sharper at what they say is a lack of substance in the Trump Kim declaration signed earlier.

"We expected it would be a flop, but it's floppier than anything we expected," says Andrei Lankov from Kookmin University in Seoul.

"A tremendous, bigly, grand nothing burger," says Van Jackson from the University of Wellington, mocking Trump's use of language.

North Korean defector Grace Jo watched coverage of the summit at the BBC's Washington bureau.

"I feel very sorry that human rights are not being discussed," she said.

"We can't separate the nuclear issue from human rights issues. They developed nuclear arms while people were dying of starvation, others in prisons."

Read more about her story here

Image source, AFP

"Nice to meet you Mr President," were the words spoken by Mr Kim upon meeting Mr Trump.

It was a firm handshake of equals pressed upon each other, but one that was significantly shorter than Mr Trump is used to, lasting only seconds instead of the uncomfortably long ones the US leader usually gives.

On Mr Kim's part, the handshake is a lot less dramatic than the one the North Korean leader gave South Korean President Moon Jae-in just a month ago, where he gripped Mr Moon's hand and led him over to the North Korean side of the border.

Read more analysis of the handshake here

BBC Monitoring
The world through its media

Image source, KCTV

North Korean state TV started at its usual time of 15:00 local time (06:00 GMT) but did not show footage from Kim and Trump's meeting.

Instead it carried a short clip featuring a female news anchor talking about Kim's visits to popular sights in Singapore, without photos or footage of his tour.

What she said matched the text of an article in Rodong Sinmun, the ruling party’s newspaper.

The article said simply that Kim "visited various sites in the city on 11 June while sojourning in the Republic of Singapore".

More from Pompeo, who seems to be enjoying his part in the Trump Kim summit.

Here's how the declaration signing unfolded. Mr Kim and Mr Trump both had things to say.