We describe an infrared transmission study of a thin layer of bulk graphite in magnetic fields up to B=34 T. Two series of absorption lines whose energy scales as sqrt[B] and B are present in the spectra and identified as contributions of massless holes at the H point and massive electrons in the vicinity of the K point, respectively. We find that the optical response of the K point electrons corresponds, over a wide range of energy and magnetic field, to a graphene bilayer with an effective interlayer coupling 2gamma_{1}, twice the value for a real graphene bilayer, which reflects the crystal ordering of bulk graphite along the c axis. The K point electrons thus behave as massive Dirac fermions with a mass enhanced twice in comparison to a true graphene bilayer.