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Planar magnesium diboride Josephson junctions are fabricated using focused helium ion beam irradiation. A single track of ion irradiation with a 30 kV He+ beam with nominal beam diameter < 0.5 nm is used to create a normal-metal barrier on a MgB2 film deposited by hybrid physical-chemical vapor deposition. Josephson coupling is observed below the critical temperature of the electrodes for a He+ doses between 8x1015/cm2 to 4x1016/cm2. Analysis of the temperature dependence of the normal resistance and critical voltage of the junctions shows highly uniform barriers with nearly ideal resistively-shunted junction behavior for higher-dose junctions, while nonequilibrium effects dominate the properties of lower-dose junctions over most of the temperature range. These results demonstrate that focused helium ion beam irradiation can produce high-quality proximity-coupled MgB2 Josephson junctions with tailorable properties, promising for use in superconducting devices and circuits. [ABSTRACT FROM AUTHOR]