Announcing the lower-level talks held Sunday, State Department spokeswoman Heather Nauert said, "We continue to prepare for a meeting between the President and North Korean leader Kim Jong Un".
The two delegations are focused on the substance of any summit between the Trump and Kim Jong Un: the issue of North Korea's nuclear weapons program.
Saturday's Korean summit came hours after South Korea expressed relief over revived talks for a Trump-Kim meeting.
Mr Kim reaffirmed his commitment to "complete" denuclearisation of the Korean Peninsula and to a planned meeting with Mr Trump, Mr Moon told a news conference in Seoul.
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The weekend meetings were the latest dramatic turn in a week of diplomatic ups and downs over the prospects for an unprecedented summit between the United States and North Korea, and the strongest sign yet that the leaders of the two Koreas are trying to keep the meeting on track.
In previous failed talks, North Korea said it could consider giving up its arsenal if Washington removed its troops from South Korea and withdrew its so-called nuclear umbrella of deterrence from South Korea and Japan.
The Moon and Trump administrations very much want the meeting, he said, "and Kim says he wants a summit, so it's going to happen". Randall Schriver, the assistant secretary of defense for East Asia and one of the officials who accompanied Secretary of State Mike Pompeo to Pyongyang earlier this month, also is in Seoul at the moment.
"I think some things need to be decided before we have that meeting", Blunt said on "Fox News Sunday". "It will happen!" he tweeted. "We will soon see where it will lead, hopefully to long and enduring prosperity and peace".
Rubio said he was extremely skeptical that Kim might actually give up his nuclear arsenal.
"A meeting was held because officials of both countries thought that meeting face-to-face would be better than a phone call", Moon said on Sunday.
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"I support the letter that President Trump sent to Kim Jong Un", Clapper said.
A three-way summit was proposed by Moon and Kim when they first met at the border village of Panmunjom on April 27, Yonhap said.
"We've got the South Korean president, I believe, providing a great benefit in this discussion, unless we are not all communicating in the same way", said Sen.
Moon is the only South Korean leader to have met a North Korean leader twice, both times in the DMZ, which is a symbol of the unending hostilities between the nations after the Korean War ended in 1953 in a truce, not a peace treaty.
Galston said Americans were unlikely to blame Trump for the summit cancellation, even as they would have credited him had it proceeded with success.
-North Korea summit in June, government officials said Sunday.
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