Moon and Kim met for about two hours in a building on the North Korean side of the truce village of Panmunjom in the heavily fortified border that divides the peninsula, spokesman Yoon Young-can said.
South Korean President Moon Jae-in bids farewell to North Korean leader Kim Jong Un as he leaves after their summit at the truce village of Panmunjom, North Korea, in this handout picture provided by the Presidential Blue House on Saturday.
Moon has played a key role in arranging the proposed meeting between Trump and Kim, and he expressed relief after the USA president suggested the June 12 summit - which he canceled in a letter to Kim on Thursday - could be back on.
On Thursday, President Trump cancelled the planned summit after a threatening statement issued by North Korea.
But White House spokeswoman Sarah Sanders said an advance team of White House and U.S. State Department officials would leave for Singapore on schedule this weekend to prepare for a possible summit there.
Trump, in his letter to Kim, said the "tremendous anger and open hostility" recently shown by North Korea led him to conclude that it was not the appropriate time to hold talks with Kim.
The President will announce the outcome of the talks on Sunday morning, he said.
Photos and footage of the meeting released by the presidential palace showed that Moon was accompanied by Suh Hoon, the chief of South Korea's National Intelligence Service spy agency.
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North Korea has conducted six nuclear tests since 2006 and numerous tests of intercontinental ballistic missiles that could, theoretically, reach the US.
Trump, who now suggests there's still a chance of holding a summit, blamed Kim's trip to China for creating an unwelcome "change in attitude" by the North Korean leader, a view shared by some observers.
Earlier on Friday, Trump told reporters the United States and North Korea still wanted to arrange a landmark meeting between the two leaders.
However, the USA president later tweeted that the summit could be back on and held in Singapore as originally planned - with Seoul expressing relief that the prospect of talks had been revived. "I believe one of the keys to the peace and stability of the peninsula is the US-DPRK relations, and a summit between them is essential".
Trump's decision to pull out of the summit with Kim came just days after he hosted Moon in a White House meeting in which he openly cast doubts on the Singapore meeting but offered no support for continued inter-Korean progress, essentially ignoring the North's recent attempts to coerce the South. "We'll see what happens - it could even be the 12th".
Trump's administration cited numerous factors in canceling the summit. DPRK stands for the North's official name, the Democratic People's Republic of Korea.
North Korea's vice foreign minister said his country remained open to resolving issues with Washington "regardless of ways, at any time, any format".
USA defense and intelligence officials have repeatedly assessed the North to be on the threshold of having the capability to strike anywhere in the continental US with a nuclear-tipped missile-a capacity that Trump and other US officials have said they would not tolerate.