The novel instrument measures the response to compression and shear of a class of materials we call soft compressible solids, examples of which are textile solids and foams. The sample is fixed between two parallel plates, and deformation is controlled by an x,y,z displacement on one plate while the stress is measured on the other. A key component is the triaxial stress transducer, which eliminates edge effects by only measuring the stress on an interior region of the plate. The edge effects and associated measurement errors are analyzed by simple isotropic elasticity theory, and compared to measurements done on chloroprene rubber foam.