For any $T \geq 1$, there are constants $R=R(T) \geq 1$ and $ζ=ζ(T)>0$ and a randomized algorithm that takes as input an integer $n$ and two strings $x,y$ of length at most $n$, and runs in time $O(n^{1+\frac{1}{T}})$ and outputs an upper bound $U$ on the edit distance $ED(x,y)$ that with high probability, satisfies $U \leq R(ED(x,y)+n^{1-ζ})$. In particular, on any input with $ED(x,y) \geq n^{1-ζ}$ the algorithm outputs a constant factor approximation with high probability. A similar result has been proven independently by Brakensiek and Rubinstein (2019).