A huge scandal erupted in 2014 when French glossy magazine Closer said Hollande, who then lived with his partner Valerie Trierweiler, routinely hopped on a scooter to spend the night with his 41-year-old mistress, publishing pictures of the three-wheeler.
Hollande went on to marry Gayet in 2022, and on Sunday, France's most famous scooter was auctioned off at a château in the town of Montbazon in western France, according to an AFP correspondent.
The auction, which was organised by the Rouillac auction house and attended by around 40 people, lasted just 10 minutes.
The grey Piaggio MP3 125 was purchased by Denis Breheret, owner of a car museum.
Breheret wants to exhibit the scooter at his museum in the town of Jallais so that "everyone can enjoy it".
The scooter "is part of French history," he said.
"I will be presenting it among the 120 vehicles that I already own, which range from 1922 to the present day."
The Elysee Palace, which was the first owner of the scooter, sold it in 2015.
The starting price for the vehicle at the auction on Sunday was 10,000 euros.
"It's the Rolls (Royce) of scooters, but its value on the second-hand market is only estimated at between 1,300 euros and 4,000 euros", said auctioneer Aymeric Rouillac.
"In January 2014, François Hollande's scooter entered the category of legendary vehicles," the Rouillac auction house said in a statement.
"It tells the story of a man like any other and of a nation torn between its desire for power and the realisation of its decline."
Trierweiler took revenge on Hollande for spurning her for Gayet with a sensational 2014 kiss-and-tell memoir called "Thank You for This Moment", which all but sank Hollande's presidency.
Before his relationship with the journalist Trierweiler, Hollande was involved for three decades with Segolene Royal, a French Socialist politician who unsuccessfully ran for president in 2007. They have four children.
Speaking on television in April, a smiling Hollande, 69, said: "I've always had scooters, and this was the last one."
