The notion of fertility in word alignment (the number of words emitted by a single state) is useful but difficult to model. Initial attempts at modeling fertility used heuristic search methods. Recent approaches instead use more principled approximate inference techniques such as Gibbs sampling for parameter estimation. Yet in practice we also need the single best alignment, which is difficult to find using Gibbs. Building on recent advances in dual decomposition, this paper introduces an exact algorithm for finding the single best alignment with a fertility HMM. Finding the best alignment appears important, as this model leads to a substantial improvement in alignment quality.