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Magnesium diboride has many intriguing characteristics, including its relatively high critical temperature and two-band nature. Most prior studies of MgB2 thin film Josephson junctions have been conducted above 2 Kelvin. We report results of sub-1 Kelvin experiments of MgB2/insulator/Pb junctions whose a-b plane is exposed for electron tunneling. By measuring differential conductance at low temperature, new details in the structure of the sigma- and pi-band gaps are observed in this data, consistent with theoretical predictions.