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Consider a bipartite graph with a set of left-vertices and a set of right-vertices. All the edges adjacent to the same left-vertex have the same weight. We present an algorithm that, given the set of right-vertices and the number of left-vertices, processes a uniformly random permutation of the left-vertices, one left-vertex at a time. In processing a particular left-vertex, the algorithm either permanently matches the left-vertex to a thus-far unmatched right-vertex, or decides never to match the left-vertex. The weight of the matching returned by our algorithm is within a constant factor of that of a maximum weight matching.