In the light of the tremendous progress that has been made in raising the transition temperature of the copper oxide superconductors (for a review, see ref. 1), it is natural to wonder how high the transition temperature, Tc, can be pushed in other classes of materials. At present, the highest reported values of Tc for non-copper-oxide bulk superconductivity are 33K in electron-doped CsxRbyC60 (ref. 2), and 30K in Ba1-xKxBiO3 (ref. 3). (Hole-doped C60 was recently found to be superconducting with a Tc as high as 52K, although the nature of the experiment meant that the supercurrents were confined to the surface of the C60 crystal, rather than probing the bulk.) Here we report the discovery of bulk superconductivity in magnesium diboride, MgB2. Magnetization and resistivity measurements establish a transition temperature of 39K, which we believe to be the highest yet determined for a non-copper-oxide bulk superconductor.