We experimentally studied the Josephson supercurrent in Nb/InN-nanowire/Nb junctions. Large critical currents up to 5.7 $mu$A have been achieved, which proves the good coupling of the nanowire to the superconductor. The effect of a magnetic field perpendicular to the plane of the Josephson junction on the critical current has been studied. The observed monotonous decrease of the critical current with magnetic field is explained by the magnetic pair-breaking effect in planar Josephson junctions of ultra-narrow width [J. C. Cuevas and F. S. Bergeret, Phys. Rev. Lett. 99, 217002 (2007)]
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