An experimental study on the vibration of a payload isolated through a quasi-zero stiffness (QZS) isolator subject to vertical ground motion is presented. QZS isolators exploit kinematic nonlinearities to enhance the vibration attenuation of a linear mass-spring system. In this paper, stepped-sine tests have been performed to identify the system parameters and determine the isolator transmissibility. The parameter identification has been carried out by fitting a phenomenological model to the experimental data. The QZS isolator transmissibility exhibits a natural frequency reduction, and the system effectively suppresses the ground harmonic vibration about the fundamental frequency of the mass-spring isolator. A non-negligible presence of dry friction has been observed, leading to the stick-slip phenomenon, and affecting the suspension activation at low frequencies. Then, the suspension response under different realistic earthquake signals is presented and critically discussed.