Finite-element methods (FEM) have been employed to calculate the stresses set up in a thin rubber layer, bonded between two rigid spheres, when small tensile or compressive deflections are imposed. Values of stiffness have been calculated for various spacings of the spheres, i.e., for various thicknesses of the rubber layer. They are in good agreement with earlier experimental measurements of compression stiffness and with the predictions of an approximate theoretical treatment. However, they are strongly affected by small departures of the rubber layer from complete incompressibility. The highest dilatant stress (in tension) is found to be set up on the central axis, near the bonded surfaces, where internal failure has been observed to occur in similar bonded layers. For moderately thick layers, the axial tensile stress in the center of the layer is substantially higher than the lateral stresses. This feature suggests that the initiation of failure in this location may not obey the same criterion as for an isotropic stress field, and that a crack, once formed here, will propagate as a tear across the axis of symmetry.