We show how to compute the edit distance between two strings of length n up to a factor of 2Õ ( √ log n) in n 1+o(1) time. This is the first sub-polynomial approximation algorithm for this problem that runs in near-linear time, improving on the state-of-the-art n 1/3+o(1) approximation. Previously, approximation of 2Õ ( √ log n) was known only for embedding edit distance into ℓ1, and it is not known if that embedding can be computed in less than a quadratic time.