The tragedy was one the deadliest mountaineering accidents to hit the Himalayan nation in the past few years. "Nine climbers were taken forever by a snow storm but their bravery and tireless spirit demonstrated by their attempt to find a new route will not be buried with them", Moon's statement said.
The group, five South Koreans and their four Nepali guides, were trying to ascend Mount Gurja, a 7,193m (23,599ft) peak.
A police team heading toward the base camp on foot is unlikely to reach the site until Sunday.
In a Facebook post, South Korean President Moon Jae-in identified the South Korean climbers as team leader Kim Chang-ho and team members Lee Jae-hoon, Yim Il-jin, Yoo Young-jik and Jung Joon-mo.
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The previous year, an avalanche above Everest's base camp killed 16 Nepalese Sherpa guides.
In 2013, the expedition's leader, Chang-ho, set a record as the fastest climber to reach the peaks of the world's 14 tallest mountains without using supplemental oxygen, mountaineering officials say, in ascents made since 2005. Compare that to Mt. Everest, which more than 4,800 climbers have summited - many of them sherpa who have achieved the feat a couple of times.
The South Korean ministry told reporters the storm blew the victims from their base camp off a steep cliff.
A rescue helicopter confirmed seeing bodies scattered in the ruins of the camp but were unable to retrieve them due to the strong winds and icy conditions.
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The bodies were brought to Nepal's Pokhara airport and flown to Kathmandu TU Teaching hospital by helicopter.
An official from the Corean Alpine Club said the families of the deceased climbers and its club members originally planned to go to Nepal and cremate the bodies before transporting them to South Korea for funeral.
Thousands of climbers flock to Nepal each year - home to eight of the world's 14 highest peaks - creating a lucrative mountain tourism industry that is a vital source of cash for the impoverished country.
"Rescuers will try to recover the bodies on Sunday", the official said.
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A helicopter rescue team spotted all the bodies amid the wreckage of the camp on Saturday morning but strong winds hampered the effort to retrieve the bodies.
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