The contest is being held in honour of the DVD release of his most recent album Sea Change, says NME.com.
The package ties a different video to each of the tracks on the album, and the rabbit, Rocket, appears in the clip for Lonesome Tears.
Fans willing to offer Rocket a good home are being asked to write an essay explaining why Beck should allow Rocket into their custody.
Beck will choose the winner himself, and four runners up will receive signed copies of the Sea Change DVD.
Full details of how to enter can be found on Beck’s official website www.beck.com.
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