We report on measurements of the low-temperature specific heat Cp(T, B) of the unconventional superconductor UBe13 in magnetic fields up to 7 T. At B≈2 T, a substantial change in the magnetic-field dependence of the temperature derivative of the magnetic-field induced contribution to the electronic specific heat CH(T, B), resulting from the flow of supercurrents around the vortices, is observed, suggesting a crossover between two different regions in the superconducting phase diagram of UBe13. For fields B >2 T, CH(T, B) exhibits a scaling behavior with respect to TB-1/2, which provides substantial evidence for the existence of point nodes in the quasiparticle excitation spectrum of the superconductor.