We study the following Constrained Bipartite Edge Coloring problem: We are given a bipartite graph G=(U,V,E) of maximum degree l with n vertices, in which some of the edges have been legally colored with c colors. We wish to complete the coloring of the edges of G minimizing the total number of colors used. The problem has been proved to be NP-hard even for bipartite graphs of maximum degree three. Two special cases of the problem have been previously considered and tight upper and ower bounds on the optimal number of colors were proved. The upper bounds led to -approximation algorithms for both problems. In this paper we present a randomized (1.37+o(1))-approximation algorithm for the general problem in the case where max{l,c}=ω(ln n). Our techniques are motivated by recent works on the Circular Arc Coloring problem and are essentially different and simpler than the existing ones.