South Korea later said the two men agreed to a second Kim-Trump summit "at the earliest possible date".
Pompeo and Kim met for about two hours and then had lunch together.
But a summit of the leaders of North and South Korea last month has helped to rekindle the peace process, as has the apparent desire of both Kim and Trump to meet again for a second summit.
The two had "refined options" for a location and date of a second summit without offering specifics, the State Department statement said.
Absent sanctions relief, North Korea has rejected turning over what Washington really wants in return: a complete inventory of its nuclear and ballistic missile facilities that could be used by global inspectors to verify they have been dismantled.
Kim also praised their "nice meeting", telling Pompeo via an interpreter following the morning's talks that it was "a very nice day that promises a good future. for both countries".
North Korea's state newspaper, Rodong Sinmun, featured eight photos of the meeting on its front page, including shots of Kim and Pompeo smiling and shaking hands, as well as some with Kim's sister Kim Yo Jong.
"Secretary Pompeo said he and Chairman Kim concurred that they will hold the second U.S".
Last month the North's foreign minister told the United Nations there was "no way" his country would disarm first as long as tough USA sanctions remain against his country. US officials have thus far said sanctions will remain in place until the North's denuclearization is fully verified. "Progress made on Singapore Summit Agreements! After having a nice meeting we can enjoy a meal together", Kim told Pompeo as they walked down a hallway toward a guesthouse dining room.
"I have asked Secretary of State Mike Pompeo not to go to North Korea, at this time, because I feel we are not making sufficient progress with respect to the denuclearization of the Korean Peninsula", Trump tweeted in August. "And we had a very successful morning, so thank you and I am looking forward to our time here at lunch as well", Pompeo said.
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Stephen Biegun, the U.S. Special Representative for North Korea, Kim Yo-jong, the North Korean leader's younger sister, and Kim Song-hye, a senior North Korean official in charge of reunification, were seen behind Pompeo and Kim in the photo. President Trump sends his regards.
"If you do see concrete action that assures the USA, the rest of the world that definitely concrete steps are being taken to eliminate very important parts of their nuclear program, then that's a definite step forward and builds the trust", Kang told The Washington Post in an interview last week. On Wednesday, he said there was unanimous support for this at last week's U.N. General Assembly, even if Russian Federation and China "had some ideas about how we might begin to think about a time when it would be appropriate to reduce them".
Pompeo's trip to Pyongyang, his fourth this year, followed a stalemate as North Korea resisted Washington's demands for irreversible steps to give up its nuclear arsenal, including a complete inventory of its weapons and facilities. He came away from it saying the two sides had made progress, only for North Korea to denounce him for making "gangster-like" demands and raising "cancerous" issues.
Nauert also said she could not comment much further on the goal of Pompeo's time in China.
Japan has been wary of Trump's initiative, fearing it could affect its long-standing security relationship with the U.S.
North and South Korea want a "phased" approach with Pyongyang rewarded as it it happens.
Pompeo will be in Tokyo Saturday and Sunday and has already met with Japanese Prime Minister Shinzo Abe there.
Both Koreas have differed with the United States on North Korea's nuclear program.
Nauert also could not name who Pompeo would be meeting with in Beijing - just that "there's the expectation that he meets with someone who's his appropriate counterpart".