Low-energy quasiparticle excitations in the superconducting (SC) state of UBe_{13} were studied by means of specific-heat (C) measurements in a rotating field. Quite unexpectedly, the magnetic-field dependence of C(H) is linear in H with no angular dependence at low fields in the SC state, implying that the gap is fully open over the Fermi surfaces, in stark contrast to previous expectations. In addition, a characteristic cubic anisotropy of C(H) was observed above 2 T with a maximum (minimum) for H∥[001] ([111]) within the (11[over ¯]0) plane, in the normal as well as in the SC states. This oscillation possibly originates from the anisotropic response of the heavy quasiparticle bands, and might be a key to understand the unusual properties of UBe_{13}.