The online retailer did not reveal how much it paid for the service, which combines online rentals on PCs, internet-enabled TVs and the Playstation 3 with offline rentals by post.
Amazon already owns a significant majority sharehgolding in LoveFilm, which operates in the UK, Germany, Sweden, Norway and Denmark, but does not operate any similar business itself in Europe.
"LoveFilm has been innovating on behalf of movie rental customers across Europe for many years, and with the advent of the LoveFilm player, they are further delighting customers by streaming digital movies for their immediate enjoyment," said Greg Greeley, Amazon's vice-president of European retail.
"LoveFilm and Amazon have enjoyed a strong working relationship since LoveFilm acquired Amazon Europe's DVD rental business in 2008, and we look forward to a productive and innovative future."
Simon Calver, chief executive of LoveFilm International, added that the deal ensures the company can "significantly enhance our members' experience across Europe".
The deal is expected to be completed by the end of the first quarter, but will first have to be approved by regulators.