Fisher matrix and related studies have suggested that, with second-generation gravitational-wave detectors, it may be possible to infer the equation of state of neutron stars using tidal effects in a binary inspiral. Here, we present the first fully Bayesian investigation of this problem. We simulate a realistic data analysis setting by performing a series of numerical experiments of binary neutron-star signals hidden in detector noise, assuming the projected final design sensitivity of the Advanced LIGO-Virgo network. With an astrophysical distribution of events (in particular, uniform in comoving volume), we find that only a few tens of detections will be required to arrive at strong constraints, even for some of the softest equations of state in the literature. Thus, direct gravitational-wave detection will provide a unique probe of neutron-star structure.