The move has surprised the industry because Redstone had publicly admitted his frustration at missed opportunities for Viacom to acquire other sites, including MySpace -- bought by News Corporation for $580m.
Viacom had a chance to buy MySpace for $500m but did not take it up. Redstone has described losing the deal as "humiliating", and it led directly to the departure of CEO Tom Freston in September.
The media giant's withdrawal from the race will lead to speculation that its digital strategy is confused.
The price is almost double the $530m News Corp paid for MySpace and is the success of that site that has pushed Facebook so much higher.
In a television interview, Redstone said that Viacom was looking for something at a less developed stage than Facebook.
He said: "We would look at companies on the cutting-edge of greatness and we could make it grow."
In the same interview, he would not say whether Viacom was planning to buy YouTube, another community-driven web phenomenon. However, he described it as a "very good company".
Facebook started as a networking site for schools and colleges but is now available for any user to sign up to.
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