The discovery of superconductivity at 39K 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 Nb sub(3)Sn 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 7T at liquid helium temperature (4.2K), significantly lower than about 10T for Nb-Ti (ref. 6) and similar to 20T for Nb sub(3)Sn (ref. 7). Here we show that MgB sub(2) 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 4.2K greater than 14T. In addition, very high critical current densities at 4.2K are achieved: 1MAcm super(-2) at 1T and 10 super(5)Acm super(-2) at 10T. These results demonstrate that MgB sub(2) has potential for high-field superconducting applications.