We show how to determine whether the edit distance between two given strings is small in sublinear time. Specifically, we present a test which, given two n-character strings A and B, runs in time o(n) and with high probability returns ``CLOSE'' if their edit distance is O(n^\alpha), and ``FAR'' if their edit distance is Omega(n), where \alpha is a fixed parameter less than 1. Our algorithm for testing the edit distance works by recursively subdividing the strings A and B into smaller substrings and looking for pairs of substrings in A, B with small edit distance. To do this, we query both strings at random places using a special technique for economizing on the samples which does not pick the samples independently and provides better query and overall complexity. As a result, our test runs in time n^max{\alpha/2, 2\alpha - 1}$ for any fixed \alpha < 1. Our algorithm thus provides a trade-off between accuracy and efficiency that is particularly useful when the input data is very large. We also show a lower bound of Omega(n^{\alpha/2}) on the query complexity of every algorithm that distinguishes pairs of strings with edit distance at most n^\alpha from those with edit distance at least~n/6.