We report low-temperature measurements of the specific heat and thermal expansion on a Ube13 single crystal and several U1-xThxBe13 polycrystals (x ≤ 0.03). A new ”line of anomalies”, B*(T), located in the B-T phase diagram below BC2(T) is discovered for the heavy-fermion superconductor UBe13. B'(T) starts at TL ≈ 0.7 K (B = 0) and terminates at B*(O), ≾ 5T. Very similar anomalies are also found for U1-xThxBe13 with x = 0.01 and 0.017. A new ”line of anomalies”, TL(x), is established in the T - x phase diagram. It hits the multi-critical point at xcr ≈ 0.019 and merges smoothly into the Tc2(x) line that marks the lower of the two phase transition temperatures at x > xcr. The implications of our results for the understanding of the low-temperature phases of pure and thoriated Ube13 are discussed.