We report the first measurements of the magnetic-field penetration depth λ in the heavy electron superconductor UBe13, performed using a SQUID magnetometer. We find the temperature dependence of λ(T)-λ(0) to follow a T2 law at low temperatures, giving further evidence of extreme gap anisotropy in this compound. We calculate the temperature dependence expected for a variety of anisotropic states, including those representing certain classes of "exotic" pairing. In general situations, the supercurrent is not parallel to the vector potential, and a more complicated field penetration takes the place of the normal Meissner effect. We argue that the data are consistent with an energy gap with point nodes on the Fermi surface but inconsistent with the large value of the Landau parameter F1S expected for a translationally invariant Fermi liquid with large effective mass. © 1986 Springer-Verlag.