A surprisingly strong variation of resistance with a perpendicular magnetic field, and a peak in the resistance versus field, R(B) has been found in insulating films of a sequence of homogeneous, quench-condensed films of amorphous Bi undergoing a thickness-tuned superconductor-insulator transition. Isotherms of magnetoresistance, rather than resistance, versus field were found to cross at a well-defined magnetic field higher than the field corresponding to the peak in R(B). For all values of B, R(T) was found to obey an Arrhenius form. At the crossover magnetic field the prefactor became equal to the quantum resistance of electron pairs h/4e2, and the activation energy returned to its zero-field value. These observations suggest that the crossover is the signature of a quantum phase transition between two distinct insulating ground states, tuned by magnetic field.