The body of Luca Lombardini was recovered by the mountain police rescue service. Those of his brother and his fiancée were located nearby but have yet to be retrieved, AFP reports.
"They are probably buried under rocks and ice in an area covering a few dozen square metres," an official at the rescue service said.
They were found around 11am UT at the foot of the Aiguille Verte ("Green Needle"), one of the peaks in the Mont Blanc massif.
"They were going along a ridge at an altitude of 3,400 to 3,500 metres, and it seems they slipped and fell together," the official said.
The ascent was a birthday present for Luca, who was joined by his fiancée and his brother Alessandro, himself a mountain rescue worker in Bardonecchia, near the French border, Italy's Ansa news agency reported.
The first body was recovered after rescuers cut the cord linking the three climbers, and attempts to recover the other two will resume Saturday morning.
The accident brings to 15 the death toll on the 4,810-metre Mont Blanc during this year's climbing season so far, after 14 people died and two went missing last year.
Overcrowding has increased the risks for the roughly 300 climbers who attempt to reach the top each day.
Officials last month began limiting access on the most popular route up Mont Blanc by turning away climbers who do not have reservations at the 120-bed Gouter refuge.
The heatwave that gripped much of Europe in recent weeks has also made conditions more treacherous, causing more ice to melt, making the ground more instable.