The man's climbing partner notified emergency services about 8.30am after the man fell near the summit ridge, on the Porter Col route of the mountain.
Police said it was believed the man may have fallen vertically some 600m.
A rescue party from the Department of Conservation's alpine rescue team responded about 10am and established the fallen man had died as a result of the fall.
The rescue team returned the man's uninjured partner shortly before midday before retrieving the man's body about half an hour later.
Police said the pair were from overseas and had gone into Aoraki Mt Cook National Park yesterday with the intention of climbing the mountain.
The men had been skiing and traversing the lower summit at about 3700m at the time of the accident.
Climbing conditions were good and the weather was fine.
The climbers have been described as very experienced and had successfully skied and climbed the east face of Aoraki Mt Cook and The Footstool the previous week.
Police said it was fortunate that members of the DoC alpine rescue team were training with land search and rescue staff in the park at the time of the accident.
The names of the climbers and their country of origin would be released later.
The climbing death is the second within a week after two climbers died on Mt Taranaki over Labour Weekend.
Nicole Sutton, 29, and her partner Hiroki Ogawa, 31, died after huddling in an ice cave for two nights when they became trapped near the summit in bad weather.
