Amazon has agreed to buy consumer film rental service Lovefilm Intl. to boost its presence in the European online video market.

Lovefilm has nearly 1.6 million members in the U.K., Germany, Sweden, Denmark and Norway, making it one of Europe’s leading subscription entertainment services.

Like Netflix, it offers DVD and videogame rental by mail and is expanding fast into streaming films and TV shows over the Internet to watch on TV, home computer or PlayStation 3 via its Lovefilm Player, which bowed in March. Amazon is expected to accelerate the growth of this business.

Amazon already holds a 42% stake in Lovefilm, after selling its DVD rental biz in the U.K. and Germany to the company in 2008.

No price has been announced for Amazon’s acquisition of the remaining shares in Lovefilm from venture capital firms Arts Alliance Media, Balderton, DJ Esprit and Index Ventures. However, analysts expect the deal, due to close this quarter, to value Lovefilm at about £200 million ($320 million).

“The deal is a winner for the members who love Lovefilm because of its value, choice, convenience and innovation in home entertainment. With Amazon’s unequivocal support, we can significantly enhance our members’ experience across Europe,” said Lovefilm chief exec Simon Calver.

“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 VP of European retail.

Lovefilm launched as a U.K. video rental service in 2004, merged with Video Island in 2006 and bought Amazon’s European subscriber base in 2008. It offers more than 70,000 titles to its subscribers, who pay up to $30.38 a month.