To explore the future clinical potential of improved soft-tissue visibility with grating-based X-ray phase contrast (PC), we have developed a first preclinical computed tomography (CT) scanner featuring a rotating gantry. The main challenge in the transition from previous bench-top systems to a preclinical scanner are phase artifacts that are caused by minimal changes in the grating alignment during gantry rotation. In this paper, we present the first experimental results from the system together with an adaptive phase recovery method that corrects for these phase artifacts. Using this method, we show that the scanner can recover quantitatively accurate Hounsfield units in attenuation and phase. Moreover, we present a first tomography scan of biological tissue with complementary information in attenuation and phase contrast. The present study hence demonstrates the feasibility of grating-based phase contrast with a rotating gantry for the first time and paves the way for future in vivo studies on small animal disease models (in the mid-term future) and human diagnostics applications (in the long-term future).