The ultrasound pulse echo overlap method has been used to determine the elastic stiffness coefficients and the corresponding ultrasonic attenuation for a single crystal of the Ni40Ti50Cu10 alloy as a function of temperature. The elastic stiffness coefficients exhibit anomalies near the martensitic phase transition. In the shear stiffness coefficient, corresponding to C44 propagation mode in austenite, a large jump occurs from 36 GPa, above the transition, down to 15 GPa, below the transition. This jump is accompanied by a strong increase in the ultrasonic attenuation. The stiffness coefficients corresponding to C11 and C' = (C11 – C12)/2 modes in austenite show an anomaly at the phase transition, however, these are small effects compared to the one associated with C44 mode. The elastic behavior of this crystal has been characterised down to a temperature of 100 K.