Pure magnesium (Mg) is an attractive metal for structural applications due to its low density, but also has low ductility and low fracture toughness. Dilute alloying of Mg with rare earth elements in small amounts improves the ductility, but the effects of alloying on fracture are not well-established. Here, the intrinsic fracture of a model Mg-3at%Y solid solution alloy is studied using a combination of anisotropic linear elastic fracture mechanics and atomistic simulations applied to a comprehensive set of crack configurations under mode I loading. The competition between brittle cleavage and ductile dislocation emission at the crack tip in Mg is improved slightly by alloying, because local fluctuations of the random solutes enable dislocation emission rather than cleavage fracture for a number of configurations where the differences in critical load for cleavage and emission are small. However, basal-plane cleavage remains strongly preferred, as in pure Mg. The alloys do show higher fracture toughness for all configurations due to local solute-induced deformation phenomena at the crack tip. Thus, alloying with Y is expected to improve the fracture toughness of Mg, but the persistence of basal cleavage prevents the alloy from becoming intrinsically ductile for all orientations.