North Korea has conducted all six of its nuclear tests at the Punggye-ri site, which consists of tunnels dug beneath Mount Mantap in the northeast of the country.
A small group of international media selected by North Korea was on hand to witness the demolition, which Pyongyang says is proof of its commitment to end nuclear testing.
The destruction of the site began at about 11am with the blowing up and collapsing of a tunnel and an observation post.
After some time, it destroyed another tunnel and other facilities, then a third tunnel and another observation post.
About five hours after beginning the operation, it destroyed two military barracks, South Korean media said in pool reports.
The North Korean offer to scrap the test site has been seen as a major concession in months of easing tension between it, on the one hand, and South Korea and the United States on the other.
The North objects to US demands that it unilaterally give up its arsenal of nuclear bombs and ballistic missiles.
Last week, Trump sought to placate North Korea after it threatened to call off the summit, saying Kim's security would be guaranteed in any deal and his country would not suffer the fate of Muammar Gaddafi's Libya, unless that could not be reached.
But the reference to the case of Libya has angered North Korea.
'DUMMY'
In a statement released by North Korean media on Thursday, Vice Foreign Minister Choe Son Hui called US Vice President Mike Pence a "political dummy" for comparing North Korea - a "nuclear weapons state" - to Libya, where Gaddafi gave up his unfinished nuclear development programme, only to be later killed by NATO-backed fighters.
"It is to be underlined, however, that in order not to follow in Libya's footstep, we paid a heavy price to build up our powerful and reliable strength that can defend ourselves and safeguard peace and security in the Korean peninsula and the region," Choe said.
Choe said the fate of the summit was "entirely" up to the United States.
"We will neither beg the US for dialogue nor take the trouble to persuade them if they do not want to sit together with us," said Choe.
China, North Korea's sole main ally, said it hoped the summit between Kim and Trump could proceed smoothly, after Trump cast further doubt over whether it would go ahead.
Both countries should cherish the opportunity and all parties should ensure that dialogue leads to a positive outcome, Chinese Foreign Ministry spokesman Lu Kang said at a regular briefing.
Trump said on Wednesday he would know next week whether the summit would take place as scheduled.
White House aides are preparing to travel to Singapore this weekend for a crucial meeting with North Korean officials to discuss the agenda and logistics for the summit, said US officials in Washington, speaking on condition of anonymity.