Mindful of previous times the North Korean regime has not followed through on promises, Pompeo said the United States is not going to take Kim at his word and will require concrete action.
South Korean presidential spokesman Kim Eui-kyeom added: "Moon told Trump that Kim said he and Trump would get along with each other, and Trump said he was looking forward to talks with Kim and there would be a very good result".
The two leaders spoke by telephone for 30 minutes on Saturday night, discussing Friday's inter-Korean summit talks in which South Korean President Moon Jae In and North Korean leader Kim Jong Un agreed on "complete denuclearization of the Korean Peninsula".
For Japan, resolving the issue of Japanese nationals abducted by North Korea in the 1970s and 1980s remains a top priority.
The nuclear site is called Punggye-ri, located in the north of North Korea where NPR's Elise Hu notes that "the Kim regime has flouted global rules six times to test nuclear devices underground".
News of Kim's shutdown pledge was met with cautious optimism, and followed the release of the "Panmunjom Declaration", a document signed by both leaders on Friday, which states, "South and North Korea confirmed the common goal of realizing, through complete denuclearization, a nuclear-free Korean Peninsula".
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The summit - only the third-ever between the leaders of the two neighbours - came ahead of Kim's planned meeting with US President Donald Trump after months of tensions sparked by North Korea's nuclear and long-range missile tests previous year. "North Korea's economic relationship is overwhelmingly with China".
A senior U.S. official said Singapore is being considered as a possible venue for the Trump-Kim summit.
On Friday, Trump said that although the location for the summit has yet to be decided, they have narrowed it down to two countries.
South Koreans were hopeful - though as NPR's Anthony Kuhn reports - skeptical. "We've seen some steps in the right direction, but we have a long way to go".
In an interview to be broadcast Sunday, new USA secretary of state Mike Pompeo, who met with Kim over the Easter weekend, told ABC's Jonathan Karl that their talks were "good conversation" about "serious matters" and Kim was "very well prepared". Tell us by voting in our poll and adding your comments below.
Last year, Pyongyang carried out its sixth nuclear test, by far its most powerful to date, and launched missiles capable of reaching the United States mainland.
But Kim promised to change the time zone back during his summit with Moon, Yoon said.