The satellite broadcaster made a 68p-a-share bid for the digital media firm, which runs online sports websites such as , , , , and .
The bid equates to a premium of 4p on its share price, which jumped from 55.5p last month on speculation that a deal was in the offing.
As well as its sports websites, 365 Media is also behind the Ritz Club Casino website. In October, the AIM-listed firm rebranded itself, changing its name from UKbetting to 365 Media Group in a move to better reflect its business.
Eric Semel, 365 Media's chief executive officer, said: "365 Media has grown from a small online sports book into a substantial online presence across Europe, with a broad reach of online gaming and betting products together with a network of leading online sports brands.
"I would like to thank all the staff who have contributed to creating the company as it now stands, without whom it would not have been possible."
Last month it launched 365 Digital Media, bringing together all of the group's online and mobile sales functions under a single brand, with Colin Elms appointed as director of advertising and sponsorship for the revamped division.
The acquisition forms part of Sky's continuing investment in new media. Last year, it paid £211m to acquire broadband group Easynet.
Earlier this week, Sky extended its Sky Anytime download service beyond web and mobile. It is to offer the new service to the 1.7m UK households with a Sky+ personal video recorder. The move challenges BT Vision and will allow viewers to download TV programmes, movies and sports highlights.
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