The temperature T dependence of the upper critical magnetic field Hc2 of the heavy-electron superconductors U1-xThxBe13 (x=0 and 2.9%) doped with various concentrations of Gd has been determined from low-frequency ac magnetic susceptibility measurements in magnetic fields up to 60 kOe. The Hc2(T) curves for U1-xGdxBe13 samples deviate from the Hc2(T) curves of UBe13 near Tc, which is consistant with the depairing of superconducting electrons via the Zeeman interaction between the spins of the superconducting electrons and the exchange field associated with the Gd spins. This suggests that UBe13 exhibits singlet superconductivity. In contrast, the Hc2(T) curves for (U0.97Th0.03)Be13 doped with Gd scale with Hc2(T) of pure (U0.97Th0.03)Be13 and do not reflect superconducting depairing by the Gd ions. These results are consistent with either strong spin-orbit scattering due to the presence of Th in UBe13, or to a qualitatively different type of superconductivity involving triplet spin pairing in (U0.97Th0.03)Be13. Measurements of the temperature dependence of Hc2 for U1-xRxBe13 compounds where R=La, Lu, and Ce for various compositions x as well as U0.985R0.015Be13 compounds for R=Tb, Dy, Ho, and Er are also presented and compared with the Gd-doped UBe13 system. The results of low-temperature specific-heat measurements of UBe13 doped with various concentrations of Gd are also discussed. © 1991 The American Physical Society.