Adsorption and dissociation of water on different oxygen- and iron-terminated hematite(0001) surfaces at monolayer coverage have been studied by density-functional theory calculations, including a Hubbard-like+U correction. We considered six possible surface terminations, including four oxygen- and two iron-terminations. Binding energy of water on these terminations can be as large as 1.0 eV. On these terminations the energy barrier for the dissociation of the molecularly adsorbed water is less than 0.3 eV, and in few cases the dissociation is even spontaneous, i.e., without any detectable barrier. Our results thus suggest that water can be adsorbed on the α-Fe2O3(0001) surface dissociatively at room temperature, as previously found by experiment. This study also presents a very first theoretical insight into the adsorption and dissociation of water on all known terminations of the hematite(0001) surface.