We report a measurement of the first four moments of the hadronic mass distribution in B --> X_c lv decays. The measurements are based on 89 million Upsilon(4S) --> BBbar events where the hadronic decay of one of the B mesons is fully reconstructed and a charged lepton from the decay of the other B meson is identified. The moments are presented for minimum lepton momenta ranging from 0.9 to 1.6 GeV in the B rest frame. It is expected that such measurements will lead to improved determinations of |V_{cb}| and |V_{ub}|.