We calculate the effect of epitaxial strain on the structure and properties of multiferroic bismuth ferrite, BiFeO3, using first-principles density functional theory. We find that, while small strains cause only quantitative changes in behavior from the bulk material, compressive strains of greater than 4% induce an isosymmetric phase transition accompanied by a dramatic change in structure. In striking contrast to the bulk rhombohedral perovskite, the highly strained structure has a c/a ratio of ~1.3 and five-coordinated Fe atoms. We predict a rotation of polarization from [111] (bulk) to nearly [001], accompanied by an increase in magnitude of ~50%, and a suppression of the magnetic ordering temperature. Intriguingly, our calculations indicate critical strain values at which the two phases might be expected to coexist.