We experimentally studied the Josephson supercurrent in Nb/InN-nanowire/Nb junctions. Large critical currents up to 5.7 µA 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)] Superconductor/normal-conductor/superconductor (SNS) junctions with a semiconductor employed as the N-weak link material offer the great advantage that here the Josephson supercurrent can be controlled by means of the field effect. 1,2 Gate-controlled superconductor/semiconductor hybrid devices such as superconducting field effect transistors 3 or split-gate structures 4 have been fabricated which find no counterpart in conventional SNS structures. In addition, the high carrier mobility attainable in semiconductors in combination with the phase-coherent Andreev reflection leads to novel unique phenomena in the magnetotransport. 5-7 Usually for these devices the semiconductor is patterned by conventional lithography. As an elegant alternative one can also directly create semiconductor nanostructures, i.e. nanowires, by epitaxial growth. 8 By using InAs nanowires connected to superconducting electrodes tunable Josephson supercurrents, supercurrent reversal, and Kondo-enhanced Andreev tunneling have been realized. [9][10][11] Among the various materials used for semiconductor nanowires InN is of particular interest for semiconductor/superconductor hybrid structures, since the surface accumulation layer in InN can provide a sufficiently low resistive contact to superconducting electrodes. 12-14 Due to almost ideal crystalline properties of InN nanowires electronic transport along the wires, contacted by normal metal electrodes, shows quantization phenomena, i.e. flux periodic magnetoconductance oscillations. 15 Furthermore, the carrier concentration in the surface electron gas is of the order of 10 13 cm −2 and thus about a factor of ten larger than in InAs. Consequently when combined with superconducting electrodes one can expect low resistive SNS junctions.Here, we report on transport studies of Nb/InNnanowire/Nb junctions. We succeeded in observing a pronounced Josephson supercurrent and a relatively large I c R N product of up to 0.44 mV. The latter factor, the critical current times the normal resistance, is an important figure of merit for Josephson junctions. We devoted special attention to the dependence of the critical current I c on an external magnetic field B, where a monotonous decrease of I c with B was found. This experimental finding is interpreted in the framework of a recent theoretical model for the proximity effect in narrow-width junctions with dimensions comparable or smaller than the magnetic length ξ B = Φ 0 /B,...