Ban, who was South Korea's foreign minister before serving 10 years as UN chief, told the UN Security Council that the "current reconciliatory atmosphere must be nurtured by continuing engagement of both South and North Korean authorities."
He said the United States can also play "a crucially important role" by engaging with North Korea, as South Korean President Moon Jae-in has suggested.
On Tuesday, the Trump administration disclosed that Vice President Mike Pence was set to hold a history-making meeting with North Korean officials during the Olympics, but Kim Jong Un's government canceled at the last minute.
Kim also sent more than 400 others to the Olympics including his younger sister, a 140-member orchestra, an all-female 229-member cheering squad and a demonstration taekwondo team. All the North Koreans became major attractions in the South because of the rarity of contacts with the isolated North, and the possibility of reduced tensions after a year of escalating rhetoric and Kim's increasingly sophisticated nuclear and ballistic missile tests.
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But he said North Korea's participation in the Olympic games in Pyeongchang has raised hopes and expectations around the world.
"I sincerely hope that this good reconciliatory move will continue through a genuine and meaningful dialogue which will address all the issues first of all the conciliatory process between the South and North leading to U.S.-North Korean talks for more meaningful and constructive denuclearization dialogue," Ban told reporters. "And that will be the process which I envision."
"We must keep alive this hard-won momentum for dialogue so that the narrow window of opportunity provided by this newly created momentum will be able to lead to a more meaningful and genuine dialogue process of reconciliation, peace and ultimate denuclearization of North Korea," Ban said. He said denuclearization would also help spur peace and stability in northeast Asia.