We calculate the emissivity for the direct URCA process in strongly magnetized, degenerate matter in neutron stars, under beta -equilibrium. We show that, if the magnetic field is large enough for protons and electrons to be confined to their ground Landau levels, the field-free threshold condition on proton concentration no longer holds, and direct URCA reactions are open for an arbitrary proton concentration. The direct URCA processs leads to an early phase of fast neutron star cooling. This circumstance can be used to constrain the initial magnetic field inside observed pulsars.