Single-beam acoustic tweezers have recently been demonstrated capable of selective three-dimensional trapping. This new contactless manipulation modality has great potential for many scientific applications. Its development as a scientific tool requires the precise calibration of its radiation force, specifically its axial component. The lack of calibration for this force is mainly due to its weak magnitude compared to competing effects like weight. We investigate an experimental method for the calibration of the axial stiffness of the radiation force by observing the axial oscillations of a trapped bead in a microgravity environment. The stiffness exhibits a linear relationship with the acoustic intensity and is of the mN/m order. Then, a predictive model, loaded with the experimental acoustic field, is compared to the measured stiffness with very good agreement, within a single amplitude coefficient. This study paves the way to the development of calibrated acoustic tweezers.