Users will have to pay an additional fee to receive the new service.
Alex Green, managing director for Amazon Channels Europe, said it was an "evolution of our service" and that "TV behaviour is changing, and Amazon has been part of that change".
Amazon already offers a similar service in the US, including premium channels such as HBO and Showtime. But this is the first time it has moved into this area in Europe.
The deal with Eurosport will also see it livestream premium sports event for the first time, including the French Open in tennis, which began this week.
JB Perrette, president and chief executive of Discovery Networks International, told the Financial Times that Amazon's enormous database would help Discovery target 10 billion TV screens around the world, up from the 1 billion it reaches at present.