Volcanic eruptions are preceded by increased magma pressures, leading to the inflation of volcanic edifices(1). Ground deformation resulting from volcano inflation can be revealed by various techniques such as spaceborne radar interferometry(2), or by strain- and tiltmeters(3). Monitoring this process in real time can provide us with useful information to forecast volcanic eruptions. In some cases, however, volcano inflation can be localized at depth with no measurable effects at the surface, and despite considerable effort(4,5) monitoring changes in volcanic interiors has proven to be difficult. Here we use the properties of ambient seismic noise recorded over an 18-month interval to show that changes in the interior of the Piton de la Fournaise volcano can be monitored continuously by measuring very small relative seismic-velocity perturbations, of the order of 0.05%. Decreases in seismic velocity a few weeks before eruptions suggest pre-eruptive inflation of the volcanic edifice, probably due to increased magma pressure. The ability to record the inflation of volcanic edifices in this fashion should improve our ability to forecast eruptions and their intensity and potential environmental impact.