Clarkson was dropped by the BBC for hitting a member of the show’s production team. BBC Director General Tony Hall said his behaviour “had crossed a line” and was impossible for the corporation to ignore.
May and Hammond plus executive producer Andy Wilman all quit the show soon after, prompting speculation that the four would look to make a rival show for another network or streaming service.
Since leaving Top Gear the trio have been hosting a live world tour which finishes in November at the O2.
Amazon said the show, which has not yet been named, represented a “significant investment” for the service although it declined to reveal any figures.
The trio will make at least three seasons of their new show and are expected to start work this autumn on its debut series.
In a statement provided by Amazon, Clarkson said: “I feel like I’ve climbed out of a biplane and into a spaceship.”
Hammond said: “Amazon? Oh yes. I have already been there. I got bitten by a bullet ant.” And May added: “We have become part of the new age of smart TV. Ironic, isn’t it?”
Jay Marine, vice president of Amazon Prime Video EU, said: “Customers told us they wanted to see the team back on screen, and we are excited to make that happen.
‘Millions of Prime members are already enjoying our ground-breaking original shows. We can’t wait to see what Jeremy, Richard, James and the team will create in what is sure to be one of the most globally anticipated shows of 2016.
“Our approach is to give programme makers creative freedom to be innovative and make the shows they want to make. This is just the start, you should expect to see more world-leading talent and the biggest shows on Prime Video.”