In order to estimate accurately the values of the elastic constants of a solid at high pressure and at an arbitrary temperatureTfrom the ultrasonic measurements of the velocities of elastic waves propagated in solids as a function of pressure at the temperatureT, it is necessary to knowa priorithe compressibility of the solid as a function of pressure at the temperatureT. However, this latter information is not always available. Hence, one has to make some kind of approximation to estimate the values of the elastic constants of solids at high pressure. The procedure developed here is more consistent than previous procedures. It requiresa prioriknowledge of the following values: the thermal expansion coefficient, its temperature derivative, the specific heat at constant pressure of a solid at one atmosphere, and the travel‐time measurements of the elastic waves propagated through the solid as a function of pressure at a temperatureTor at more than one temperature.