This paper presents a high-resolution hyperspectral image fusion algorithm based on spectral unmixing. The widely used linear observation model (with additive Gaussian noise) is combined with the linear spectral mixture model to form the data terms. The non-negativity and sum-to-one constraints, resulting from the intrinsic physical properties of the abundances (i.e., fractions of the materials contained in each pixel), are introduced to regularize the ill-posed image fusion problem. The joint fusion and unmixing problem is formulated as the minimization of a cost function with respect to the mixing matrix (which contains the spectral signatures of the pure material, referred to as endmembers), and the abundance maps, with non-negativity and sum-to-one constraints. This optimization problem is attacked with an alternating optimization strategy. The two resulting sub-problems are convex and are solved efficiently using the alternating direction method of multipliers. Simulation results, including comparisons with the state-of-the-art, document the effectiveness and competitiveness of the proposed unmixing based fusion algorithm.