The discovery of superconductivity at 39 K in magnesium diboride offers the possibility of a new class of low-cost, high-performance superconducting materials for magnets and electronic applications. This compound has twice the transition temperature of Nb3Sn and four times that of Nb-Ti alloy, and the vital prerequisite of strongly linked current flow has already been demonstrated. One possible drawback, however, is that the magnetic field at which superconductivity is destroyed is modest. Furthermore, the field which limits the range of practical applications-the irreversibility field H*(T)-is approximately 7 T at liquid helium temperature (4.2 K), significantly lower than about 10 T for Nb-Ti (ref. 6) and approximately 20 T for Nb3Sn (ref. 7). Here we show that MgB2 thin films that are alloyed with oxygen can exhibit a much steeper temperature dependence of H*(T) than is observed in bulk materials, yielding an H* value at 4.2 K greater than 14 T. In addition, very high critical current densities at 4.2 K are achieved: 1 MA cm-2 at 1 T and 105 A cm-2 at 10 T. These results demonstrate that MgB2 has potential for high-field superconducting applications.