"He gives us all the credit", Trump said of Moon later at the rally.
The Pompeo-Kim meeting was one in a series of high-level diplomatic talks over the Korean Peninsula, ahead of a potential meeting between Trump and Kim in late May.
The Punggye-ri site's closure will be done in public, Yoon Young-chan, the spokesperson for South Korean president Moon Jae-in, said in Sunday's announcement.
He also tweeted that "things are going very well, time and location" of meeting with Kim are being set.
According to the New York Times, Kim's concession comes as the dictator agreed to allow both USA and South Korean nuclear experts into his country to examine his facilities and testing sites.
But a senior USA official said Singapore was still high on the list of potential sites for the summit, whose date still remains to be established. "It's never gone this far", he said during the press conference with Merkel.
The new round of nuclear negotiations with North Korea comes after a decades-long cycle of crises, stalemates and broken promises that allowed the country the room to build a legitimate arsenal.
Both the countries have had different time zones since 2015 when North Korea changed its time to 30 minutes behind the South, created to mark the 70th anniversary of Korea's liberation from Japanese rule after World War II.
The tests also brought about crippling financial sanctions against the North by the global community.
North Korea had also halted such broadcasting in response, according to the ministry.
"What we've now got to do is not relent on the economic pressure until that goal is achieved", he said. "We had an extensive conversation on the hardest issues that face our two countries", Pompeo, who met with Kim prior to his confirmation as secretary of state, continued.
The North's Korean Central Television on Saturday dedicated its entire news program to coverage of Friday's inter-Korean summit.
In 2008, North Korea blew up a cooling tower at its nuclear power plant to show its commitment to stop making nuclear weapons.
The Korean Peninsula technically remains in a state of war because the 1950-53 Korean War was halted with an armistice, not a peace treaty. It's never gone like this. This enthusiasm for them wanting to make a deal.
In the meantime, the White House continues to prepare for an upcoming meeting between Trump and Kim.