North Korean leader Kim Jong-un has reaffirmed his commitment to "complete" denuclearisation of the Korean peninsula and to a planned meeting with US President Donald Trump, South Korean President Moon Jae-in says.
In Washington, Trump signalled that preparations for the summit with Kim were going ahead.
Moon and Kim agreed at a surprise meeting on Saturday (local time) that a possible North Korea-US summit must be held successfully, Moon told a news conference in Seoul.
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"Chairman Kim and I have agreed that the June 12 summit should be held successfully, and that our quest for the Korean peninsula's denuclearisation and a perpetual peace regime should not be halted," Moon said.
The meeting was the latest dramatic turn in a week of diplomatic ups and downs surrounding the prospects for an unprecedented summit between the US and North Korea, and the strongest sign yet that the two Korean leaders are trying to keep the on-again off-again meeting on track.
A statement from North Korea's state news agency KCNA said Kim expressed "his fixed will" on the possibility of meeting Trump as previously planned.
Moon, who returned to Seoul on Thursday morning after meeting Trump in Washington in a bid to keep the high-stakes US-North Korea summit on track, said he delivered a message of Trump's "firm will" to end the hostile relationship with North Korea and pursue bilateral economic co-operation.
While maintaining that Kim is committed to denuclearisation, Moon acknowledged Pyongyang and Washington may have differing expectations of what that means, and he urged both sides to hold working-level talks to resolve their differences.
In a letter to Kim on Thursday, Trump had said he was cancelling the planned Singapore summit, citing North Korea's "open hostility".
But on Saturday, Trump said he was still looking at a June 12 date for a summit in Singapore, adding that talks were progressing very well.
"We're doing very well in terms of the summit with North Korea," Trump said at the White House. "It's moving along very nicely. So we're looking at June 12th in Singapore. That hasn't changed. So, we'll see what happens."
A White House team will leave as scheduled for Singapore this weekend to prepare for the possible summit, a White House spokeswoman said.
The Trump administration is demanding that North Korea completely and irreversibly shutter its nuclear weapons program.
Moon said he held Saturday's impromptu meeting at Kim's request.
The KCNA statement said Kim and Moon agreed to hold high-level talks between their two nations on June 1, and to take steps to quickly implement their efforts to denuclearise the peninsula. KCNA said the leaders also agreed to meet frequently.
A South Korean presidential official also told reporters the two Koreas are also in talks over a three-way declaration of the end to 1950-53 Korean War but there has not been any agreement yet over a tripartite summit.