Xbox Live on Xbox One has been confirmed by Microsoft. Your Xbox Live Gold account and Achievements will carry over to Xbox One. Your Gamertag and Avatar will also transfer over. [1]
Xbox Live Arcade purchases will not transfer over, as the Xbox One will not be in any way backwards compatible. However, entertainment content that you've purchased, such as movies, will carry over. [2]
According to Microsoft, "Your Gold account carries over so you will have your Gold on Xbox 360 as well as Xbox One. With Smart Match, Game DVR and New Achievements, we previewed some of the major investments we're making to make the new Xbox Live the best place to play games and enjoy entertainment with your friends online. We will have even more to share in three weeks at E3." [Source: IGN Interview with Microsoft] [3]
"Your current Xbox Live Gamertag will stay with you on Xbox One if you choose to keep it, and your hard-earned Gamerscore and Achievements will indeed carry over from Xbox 360."
Xbox Live will have access to over 300,000 servers, which is larger than the entirety of the internet was during the late-'90s. [4]
Cloud Storage
All of your content will be available and stored on the cloud, meaning you can access your games, movies, and saves anywhere at anytime. [5] It is unclear whether this service will be available to non-Gold members.
Xbox Live Friend Cap
The cap for Xbox Live friends will also raise to 1000, up from the 360's limit of 100. [6]
Privacy
You'll be able to display various levels of privacy. Certain friends will only be able to see your Gamertag, while closer ones will see your full name. [7]
Subscription Price
The Xbox Live on Xbox One subscription price has not been revealed. An Xbox Live Gold Account currently costs $59.99 for 12 months.
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Close References & Sources (2)
http://www.ign.com/articles/2013/05/21/gamerscore-will-port-over-to-xbox-one
http://www.ign.com/articles/2013/05/21/microsoft-details-new-xbox-live