Elastic stability criteria are derived for homogeneous lattices under arbitrary but uniform external load. These conditions depend explicitly on the applied stress and reduce, in the limit of vanishing load, to the criteria due to Born, involving only the elastic constants of the crystal. By demonstrating the validity of our results through a comparison of the analysis of an fcc lattice under hydrostatic tension with direct molecular-dynamics simulation, we show that crystal stability under stress (ideal strength) is not a question only of material property, and that even qualitative predictions require the inclusion of the effects of applied stress. General implications of our findings, as well as relevance to stability phenomena in melting, polymorphism, crack nucleation, and solid-state amorphization, are discussed.