The five independent moduli required to construct the complete monocrystal elastic modulus tensor of the hexagonal-symmetry superhard compound ReB(2) were measured from 308 to 5 K using resonant ultrasound spectroscopy on a special-texture polycrystal. This is possible because, confirmed by X-ray diffraction, the specimen measured was composed of grains with hexagonal axes parallel so that its polycrystal elastic response is identical to a monocrystal and because hexagonal-symmetry solids are elastically isotropic in the plane perpendicular to the hexagonal axis. Along the hexagonal (c) axis, C(33) (0) = 1021 GPa, nearly equal to C(11) of diamond, and consistent with the superhard properties. However, in the (softer) isotropic plane, C(11) (0) = 671 GPa, much lower than diamond. The changes of C(ij) with temperature are very small and smooth. The Debye temperature was computed to be 738 K, and using a high-temperature approximation, the Grüneisen parameter is γ = 1.7.