The authors present data on the magnetic field dependence of critical currents in epitaxial films of Y1Ba2Cu3O7-δ containing a single boundary. The data are obtained as a function of temperature, the orientation of the grain boundary, and magnetic fields of up to 5 T. A significant residual critical current is observed which increases with decreasing angle of misorientation at a given field and temperature. The data are in qualitative accord with a model in which the grain boundary comprises a large number of microbridges in parallel.