Heat capacity measurements were made on a single-crystal sample of AuZn using semi-adiabatic calorimetry from T = 0.6 K to T = 100 K. This material exhibits a continuous phase transition at 64.75 K, and analysis of the heat capacity data give values for the entropy of transition (Delta(tr)S) 2.02 J . K-1 . mol(-1), electronic heat capacity (gamma) 0.38 mJ . K-2 . mol(-1) and low-temperature limiting Debye temperature (Theta(D)) 219 K. Mechanical tests performed at liquid helium temperatures verify that AuZn exhibits the shape memory effect. To the best of our knowledge, the martensitic transformation temperature in AuZn is the lowest observed for a shape-memory alloy and is the first to exhibit a continuous transition at a martensitic transition. We also report heat capacity measurements on a high-purity single crystal of copper to demonstrate the accuracy and precision of our low-temperature heat capacity apparatus.