As announced in the Amazon Prime Video iOS app release notes, the Apple TV Amazon Prime Video app is now rolling out. The release notes say users have to download a separate tvOS app, which apparently will work on the third-generation Apple TV as well.
At long last, Amazon’s Prime Video application will be available today for the tvOS-powered Apple TV, as expected based on hints from the App Store. Apple originally teased the release back at WWDC, but both companies had gone quiet on a specific launch date until now. Search ‘Amazon Prime Video’ in the Apple TV App Store to find it.