We report the use of the 9Be nuclear magnetic resonance to study heavy-fermion superconductivity in the U1-xThxBe13 alloy system, x = 0 and 0.033. The nuclear spin-lattice relaxation rate 1/T1, which yields information on thermal excitations in the superconducting state, is found to vary more slowly the temperature than expected for a conventional BCS superconductor with a nonzero energy gap Δ. This indicates an enhanced density of excitations for low energies E ≪ Δ. At intermediate temperatures 1/T1T3≃const. for UBe13, which is consistent with highly anisotropic pairing; the specific temperature dependence suggests lines of zeros of Δ on the Fermi surface. The relaxation data do not agree qualitatively with theories of superconducting pair breaking due to paramagnetic impurities. At low temperatures (T < 0.2 K) 1 T1T is approximately constant in UBe13, with a value which decreases with decreasing applied field. This behavior is probably not due to direct relaxation by paramagnetic impurities, for which the temperature dependence would be in the opposite sense. For x= 0.033 1/T1 varies less rapidly than T3. The additional relaxation may be due to filling in the gap by Th-induced pair breaking. The 9Be spectra give no indication of additional broadening due to vortex-lattice inhomogeneity or (for x = 0033) of magnetic ordering at a second transition temperature Tc2 below the superconducting transition (at Tc1). Our results are ompared to NMR studies of CeCu2Si2, and to other experiments in heavy-fermion superconductors. © 1985.