The complete set of 21 elastic stiffnesses of a composite material are found from ultrasonic measurements of the phase velocity anisotropy. Quasi-P and quasi-S wave phase speeds at a variety of incident angles within a number of differing planes through the material are obtained using the τ-p plane wave decomposition technique. The 779 individual phase velocities were inverted, under no a priori presumptions about the symmetry or orientation of the material, to provide all the stiffnesses. These show that the material has nearly orthorhombic symmetry, as is expected from its texture. This orthorhombic character was further apparent in a number of bootstrap tests of the inversion that assumed differing levels of symmetry from triclinic to orthorhombic and using various subsets of the measured phase velocities. However, the present analysis does not account for the effects of wave speed dispersion evident in the observed waveforms. This dispersion is particularly severe for the in-plane q-S polarization and is possibly a consequence of the fine-layered structure of the material.