"Some say that we are terminating facilities that are not functioning, but you will see that we have two more tunnels that are bigger than the existing ones and that they are in good condition," Moon's chief press secretary, Yoon Young Chan, quoted Kim as saying.
There have been reports that the test site, buried under Mount Mantap, was suffering from "tired mountain syndrome" and was unusable after September's huge test, which caused an earthquake so big that satellites caught images of the mountain above the site actually moving.
But numerous nuclear experts have cast doubt on that theory, and Kim apparently did, too.
Kim said he would invite security experts and journalists to the North to observe the closure of the site, Yoon said.
In Washington, National Security Adviser John Bolton said the Trump Admnistration isn't "starry-eyed" about Kim's promises. The United States, he said on Fox News, isn't ready to ease sanctions or offer other concessions to North Korea before Pyongyang fully commits to denuclearisation.
The White House, though, continues to prepare for an upcoming meeting between Trump and Kim, and Bolton said that the details are being negotiated.
"We need to agree on a place, and that remains an issue," he said. "But if, in fact, Kim has made a strategic decision to give up his entire nuclear weapons programme, then I think deciding on the place and the date should be fairly easy."
There are other issues, Bolton said, that the Administration wants to press, if not immediately, soon: "ballistic missiles, chemical and biological weapons, the American hostages, the Japanese abductees."
Secretary of State Mike Pompeo, in an interview with ABC, also brought up the issue of three Americans who are being held by North Korea.
Pompeo, who secretly met Kim in North Korea in March, said while on a visit to Israel today that Kim would not be alarmed about American intentions even if the Trump Administration withdrew from the Iran nuclear deal.
This is not the first time North Korea has invited outside experts to witness the shutting down of some aspect of its nuclear programme.
In 2008 Pyongyang invited international journalists to film the destruction of the cooling tower at the Yongbyon nuclear reactor, from which it had been harvesting plutonium to make its first bombs.
All the while, it turned out North Korea was building a separate uranium enrichment facility so it could continue to produce fissile material even without Yongbyon.
Kim reportedly said while meeting Moon that he had no intention of using his nuclear weapons against neighbouring countries.
"Although I am inherently resistant toward America, people will see that I am not the kind of person who fires nukes at South Korea, the Pacific or America," Kim said during the summit, Yoon told reporters in Seoul.
"Why would we keep nuclear weapons and live in a difficult condition if we often meet with Americans to build trust and they promise us to end the war and not to invade us?" Yoon quoted Kim as saying.
That will certainly be viewed as disingenuous, to say the least, given that Kim's representatives and state media outlets repeatedly threatened last year to fire nuclear-tipped missiles at the United States and to detonate a hydrogen bomb over the Pacific.
But this is a new year, and Kim, in a strong position having obtained demonstrably functional nuclear weapons and missiles, appears ready to deal.
South Korea's progressive president wants to use his summit with Kim as a springboard to improve Pyongyang's relations with Tokyo and, particularly, with Washington.
Moon and US President Donald Trump spoke on the phone for 75 minutes at the weekend, and agreed that South Korea and the United States should continue to closely coordinate "so that the planned US-North Korea summit generates an agreement on concrete measures to realise complete denuclearisation," the Blue House spokesman said.
Trump tweeted afterward that he "had a long and very good talk with President Moon of South Korea."
"Things are going very well, time and location of meeting with North Korea is being set," Trump said. "Also spoke to Prime Minister Abe of Japan to inform him of the ongoing negotiations."
Moon also spoke to Abe over the weekend and "offered to lay a bridge between North Korea and Japan," another Blue House spokesman said.
Moon and Chinese Premier Li Keqiang are expected to meet in Tokyo for a trilateral meeting with Abe - itself a significant breakthrough in the frosty relations in the region - on May 9.
Moon will then travel to Washington for a meeting with Trump about the latter's summit with Kim, expected to take place at the end of May or beginning of June.
Influential members of Congress expressed some doubts about relations with North Korea.
"A lot of what they are agreeing to now they have agreed to in the past. And as it has turned out, they have something very different in mind when they talk about denuclearisation," said Congressman Adam Schiff of California, the top Democrat on the House Intelligence Committee, during an appearance on ABC News.
Speaking on CNN, Senator James Lankford, R, said he would not have described Kim as "honourable," as Trump did last week after months of mocking the North Korean leader.
"I think (Trump) is better to be able to just call him 'rocket man' and to be able to stick with that than honourable just because he is a ruthless dictator that does public executions of anyone who disagrees," Lankford, a member of the Senate Intelligence Committee, said.
