We present scanning tunneling microscopy measurements of the surface of superconducting MgB2 with a critical temperature of 39 K. In zero magnetic field the conductance spectra can be analyzed in terms of the standard BCS theory with a smearing parameter gamma. The value of the superconducting gap is 5 meV at 4.2 K, with no experimentally significant variation across the surface of the sample. The temperature dependence of the gap follows the BCS form, fully consistent with phonon-mediated superconductivity in this novel superconductor. The application of a magnetic field induces strong pair breaking as seen in the conductance spectra in fields up to 6 T.