Graham Paterson, 60, who ran Skye Hi Cuillin Mountain Guides with his wife Annie, was discovered by rescue teams on Saturday morning after being reported missing on Thursday.
Another climber raised the alarm after seeing him fall on the Cuillin Ridge and police, the RAF, mountain rescue teams from Skye and Kintail and the Rescue Dogs Association were all drafted in to help search for him.
However, the island was lashed with storms and gales and the operation had to be called off on Friday.
The search was resumed on Saturday morning, when the experienced climber's body was found.
The Mountaineering Council of Scotland has offered its sympathies to Mr Paterson's family. A spokesman said: "Our thoughts are with his family and friends at this sad time and we would like to recognise the hard work and dedication of the agencies and services who were involved in the search."
Local Skye councillor Drew Millar added: "I'm obviously very sorry to hear what has happened. I didn't know Mr Paterson personally, but my thoughts and sympathy go out to his family and friends."
Friends of Mr Paterson and his wife also posted messages of condolence on social networking sites.
Greer Ralston posted on the Skye Hi Facebook site: "We have just heard this terrible news, our thoughts are with you at this awful time."
Jim Willsher wrote: "Annie, our thoughts are with you."
Mr Paterson was understood to have gone on the hills by himself, but it remains unclear what caused his fall.
Despite the report of his accident by a fellow climber, it is believed the witness had difficulty pinpointing the exact location of the fall.
A police spokesman said: "Northern Constabulary can confirm that the body recovered on Saturday 29th December from the Cuillin Hills, Isle of Skye, has been identified as that of Graham Greig Paterson, aged 60, mountain guide, from Carbost, Isle of Skye.
"A report will be submitted to the procurator-fiscal."
Mr Paterson's website says he has been guiding walks, scrambles and climbs on the Isle of Skye and elsewhere in Scotland since 1995, and adds that his wife "shares Graham's enthusiasm for climbing on the Cuillin Ridge".
Meanwhile, the funeral of a 21-year-old who was found dead two days before Christmas after attending a rugby dance is due to take place today.
The disappearance of Dave Mackay after a night out in Tain, in the Highlands, triggered a major search-and-rescue mission after his family raised the alarm.
His body was discovered on the shoreline north-west of the Glenmorangie distillery last Sunday afternoon.
A funeral service is due to be held today at Ardross Hall, followed by a burial at Alness Cemetery.
In a notice for the service, Mr Mackay's family said a "beautiful shining light" had been cruelly taken from them.