Police say 32-year-old Magnus Kastengren fell about 600 metres (2,000 feet) yesterday while he and climbing partner Andreas Fransson were ski traversing near the lower summit of Mount Cook.
The mountain is also known by its indigenous Maori name, Aoraki.
Police say Fransson was not injured and has now left the country bound for Sweden.
Police say the pair were experienced climbers and conditions were good at the time of the accident. However, the area is not recognised as being suitable for skiing and what the men were attempting is considered by experts to be extreme.