A meeting with United States President Donald Trump is expected to follow late next month or early June.
In a sign of just how much is suddenly on the table, South Korea recently confirmed it is in talks with the North and the US about signing a treaty to end the Korean War - which halted in 1953 but never formally ended.
North Korea has dropped its demand for the departure of 28,000 US troops stationed in the South as a condition for the summits taking place.
Mr Trump used his weekend tweets to criticise his opponents who accused him of giving up too much in an effort to meet with Mr Kim. The two men had exchanged insults for about a year before, suddenly, a thaw in the relationship appeared to become real.
Mr Trump says the US has not given up anything and North Korea has agreed to denuclearisation (it has not), site closure and no more testing.
South Korea officials say Pyongyang is willing to talk about denuclearisation at the expected US-North Korea summit but has not actually made a concrete commitment to abandoning its nuclear weapons programme.
For his part, Mr Kim says North Korea no longer needs any nuclear tests, mid-range and intercontinental ballistic rocket tests. The nuclear test site in the northern area has also completed its mission. The North is indicating it no longer needs to test because it now has nuclear arms.
Caution is needed over Mr Kim's news. Pyongyang must be seeking something and Mr Kim can always go back on his word.
The interesting twist is how China will react if the two Koreas become much closer and lean towards a stronger relationship with the US.
Mr Trump will be prepared to give up something to become the US president who officially ends the Korean War. A united Korean peninsula, where American troops can remain, is a mouth-watering prospect for the US president and something to make China worry.
China finds itself on the outside of a rapidly unfolding diplomatic situation and remains wary of Mr Kim's objectives in his reaching out to his nation's two most bitterest enemies.
Any loss of prestige for China or its President Xi Jinping will be a problem for anyone viewing China as an essential element of international relations.
North Korea has provided a convenient buffer between China and having thousands of American troops on its border. Since the Korean War, alliances between China and North Korea, and the US and its ally in the South, have been unshakeable.
The best outcome for China is for the US troops to leave and the two united Koreas to lean towards China for economic and diplomatic stability.
Mr Kim is making a shrewd public relations move, seen by many as trying to ease global tensions while at the same time going into negotiations in a strong position of still having China's support. China has no other options but to support Mr Kim as a way of offsetting the influence of America.
Trade sanctions have hit North Korean people hard but have hardly dented the wealth of Mr Kim and his elite colleagues. The removal of those harsh United Nation sanctions will likely be part of any summit discussions.
What South Korea must be wary of is falling into the same trap as West Germany when the Berlin Wall was finally dismantled. The West inherited a population not used to capitalism. Large amounts of money were poured into rebuilding the East, not all of it successfully.
South Korea and the US will be aware of the traps of closer economic relations with the North. But the summit is a hopeful start to defusing one of the world's nuclear trigger points.
