This paper disentangles the relationship between COVID-19 propagation and mobility. In a theoretical model allowing mobility to be endogenously determined by the COVID-19 prevalence rate, we show that an exogenous epidemic shock has an immediate effect on mobility whereas an exogenous mobility shock influences epidemic variables with a delay. In the long run, exogenous disease contagiousness and mobility jointly shape epidemiological outcomes. The short-run theoretical result allows us to recover, empirically, the causal impacts of mobility and COVID-19 hospitalisations on each other in France. We find that hospitalisations are highly sensitive to mobility whereas mobility is little influenced by hospitalisations. In France, it seems therefore that voluntary social distancing would not have been effective to control the epidemic, in the absence of social distancing mandates.