The surface stress, which characterizes the state of stress at the surface of a macroscopic crystal, is calculated from first principles by two methods for Mo(001) to be 3.1 mRyd bohr(-2) = 2.4 J m(-2). Both methods use the energy of a fully relaxed seven-layer slab as a function of the in-plane lattice parameter a. One method uses the slope and the other the curvature at particular values of a. Fully relaxed energies give surface stress values 40% smaller than partially relaxed energies which have relaxed just a single common layer spacing. The slab is divided into bulk and surface regions with different parameters. Estimates are made of the surface region parameters including its equilibrium in-plane lattice constant, its epitaxial elastic constant, its Poisson ratio for in-plane strains and its thickness.