The paper is concerned with the analysis of a singular perturbation problem for a one-dimensional hyperbolic balance law. We investigate the limit of the sequence of entropy solutions as the singular parameter tends to zero, and its relation with the set of zeroes of the reaction function. The main contribution of the paper is to give a reasonable set of sufficient conditions on the initial data such that the sequence converges almost everywhere and, under some supplementary conditions, uniformly to a piecewise constant limit function where constant states are separated by curves moving with a prescribed (constant) speed.