The elastic response of ordered 3D soap froth, in which N layers of cells are confined between two rigid walls, is analyzed. Surface Evolver simulations are used to compute the equilibrium structure, which consists of a layer of Fejes-Toth cells at each wall and N − 2 core layers of Kelvin cells. The reference state corresponds to the plate spacing h o that achieves isotropic stress; and the foam confinement is varied by changing h. The foam is sheared in two orthogonal directions to determine elastic behavior up to the elastic limit or yield strain, which corresponds to the onset of topological transitions. The shear moduli in both directions decrease as N increases and slowly converge to the values of bulk Kelvin foam; the dependence on N is well described by a three-layer model consisting of Fejes-Toth cells and a core of Kelvin cells that deform like bulk foam. The influence of foam confinement on the elastic limit is studied. The topological transitions are compared to those in bulk Kelvin foam.