Clarkson had generated both controversy and profits for Britain's publicly-funded broadcaster. His BBC contract was not renewed after he physically attacked a member of the "Top Gear" production staff in March.
The 55-year-old had become the popular face of the programme by mixing a passion for cars with a swagger and blunt commentary that offended numerous groups over the years, including environmentalists, mental health charities and cyclists.
With "Top Gear" aired in more than 200 countries and watched by 350 million viewers worldwide, his dismissal made headlines around the world.
"I feel like I've climbed out of a bi-plane and into a spaceship," he said of his new job, and in an apparent dig at his previous employer.
Amazon Prime, a subscription service that costs $99 per year or 79 pounds in Britain, offers free delivery to customers in the hope they will shop more with the online retailer.
As a further incentive, Prime members also receive a range of streaming music, movies and TV programmes.
The service, which is estimated to have around 40 million members, is trying along with its rivals to make exclusive video content a key area of differentiation. Where Netflix has 'House of Cards' and 'Orange Is the New Black', Amazon Prime has 'Transparent' and now Clarkson's new motoring show.
Amazon said the new show, to be overseen by the trio's long time executive producer Andy Wilman, will be available globally on the subscription streaming service next year and run for three seasons. It gave no further details of the as-yet unnamed show.
"Top Gear" will continue on the BBC, where Clarkson has been replaced as lead presenter by television personality and DJ Chris Evans.
Clarkson had been on a final warning from the BBC over accusations last year that he had used racist language while filming the show.
Last October, the show also caused controversy when the crew was forced to flee Argentina after driving a Porsche 928 GT with the registration number H982 FKL. That was taken by some in Argentina to refer to the Falklands war which it fought against Britain in 1982.
