In the PROPER INTERVAL COMPLETION problem we are given a graph G and an integer k, and the task is to turn G using at most k edge additions into a proper interval graph, i.e., a graph admitting an intersection model of equal-length intervals on a line. The study of PROPER INTERVAL COMPLETION from the viewpoint of parameterized complexity has been initiated by Kaplan, Shamir, and Tarjan [SIAM J. Comput., 28(1999), pp. 1906-1922], who showed an algorithm for the problem working in O(16^k. (n + m)) time. In this paper we present an algorithm with running time k^O(k2/3) + O(nm(kn + m)), which is the first subexponential parameterized algorithm for Proper Interval Completion.