In order to resolve inconsistencies encountered in published data for SrSO, the elasticity and the phase stability of celestite has been studied using thermal diffuse scattering, high pressure powder synchrotron x-ray diffraction, Raman scattering and DFT calculations. The structure of SrSO is found to be stable up to 62 GPa at ambient temperature. The preferred values for the components of the elastic stiffness tensor have been determined using x-ray thermal diffuse scattering and are (in GPa): c₁₁ = 105, c₂₂ = 92 , c₃₃ = 114, c₄₄ = 16, c₅₅ = 31, c₆₆ = 26, c₁₂ = 40, c₁₃ = 52, c₂₃ = 37. The preferred value for the bulk modulus is K = 62(2) GPa. This work shows that thermal diffuse scattering collected at two temperatures allows the determination of the full elastic tensor of crystals with low space group symmetry.