Clarkson was fired from the BBC for physically attacking a member of Top Gear's production staff in March.
Amazon said the new show, to be overseen by the trio's long time executive producer Andy Wilman, will be available on its subscription streaming service Amazon Prime next year and run for three seasons.
It gave no further details of the as-yet unnamed show.
Clarkson, 55, was replaced as lead presenter of the globally popular Top Gear show - aired in more than 200 countries and watched by 350 million viewers worldwide - by television presenter and DJ Chris Evans.
Clarkson, who had generated both controversy and profits for Britain's publicly-funded broadcaster, had been on a final warning over accusations last year that he had used racist language while filming the show.
Last October, the show sparked a diplomatic incident between Britain and Argentina, which went to war in 1982 over the Falkland Islands.
A Top Gear television crew was forced to flee Argentina after driving a Porsche 928 GT with the registration number H982 FKL, which some people suggested could refer to the Falklands conflict.
© Thomson Reuters 2015