Total-energy calculations from first principles have been made on a seven-atomic- layer slab of Mo(001) as a function of the in-plane lattice parameter with full relaxation of the layer spacings. The energy minimum gives the equilibrium state of the slab, which contracts both in plane and out of plane between one and two per cent with respect to bulk. The energy changes under deformation from equilibrium are treated as strain energies and are fitted to a composite elastic model consisting of two surface regions and a bulk region, each with its structural and elastic parameters. These parameters are evaluated in a separate calculation for the bulk region, so that subtracting the known bulk strain energies from the total strain energy permits evaluation of the parameters of the surface region. Six deformations of the slab around equilibrium give the six elastic constants of the tetragonal surface regions. The surface region material is about two atomic layers deep, slightly prolate in its own equilibrium state, substantially elastically anisotropic compared to cubic symmetry, stable, but considerably weaker elastically and closer to instability than bulk.