Pulse-echo-overlap measurements of ultrasonic wave velocity have been used to determine the elastic-stiffness tensor components C-u and the adiabatic bulk modulus, BS, of a ferromagnetic Heusler structure Cu41Mn20Al39 at % alloy single crystal as functions of temperature in the range 14-300 K and hydrostatic pressure up to 0.2 GPa at room temperature. At 295 K the elastic stiffnesses are: C-11 = 133 GPa, C-44 = 92 GPa, C＇ (= (C-11 - C-12)/2) = 17 GPa, C-12 = 99 GPa, C-L(= C-11 + C-44 - C＇) = 205 GPa, and B-S (= C-11 - 4C＇/3) = 106 GPa. Cu41Mn20Al39 is a comparatively soft material elastically because its elastic properties are influenced strongly by magnetoelastic effects. The results of measurements of the effects of hydrostatic pressure on the ultrasonic wave velocity have been used to obtain the hydrostatic-pressure derivatives of the elastic-stiffness tenser components. At 295 K (partial derivative C-11/partial derivative P)(P = 0), (partial derivative C-44/partial derivative P)(P=0), (partial derivative C＇/partial derivative P)(P=0), (partial derivative C-12/partial derivative P)(P=0), (partial derivative C-L/partial derivative P)(P=0), and (partial derivative B-S/partial derivative P)(P=0) are 5.0 +/- 0.1, 3.0 +/- 0.1, 1.0 +/- 0.2, 3.0 +/- 0.3, 7.7 +/-0.4 and 3.7 +/- 0.4, respectively. Application of hydrostatic pressure does not induce acoustic-mode softening: the pressure derivatives (partial derivative C-IJ/partial derivative P)(P=0) and (partial derivative B-S/partial derivative P)(p=0) and the acoustic-mode Gruneisen parameters are positive. An interesting feature of the non-linear acoustic behaviour of this alloy is that the value obtained for (partial derivative C＇/partial derivative P)(P=0), associated with the softer shear mode propagated a long the [110] direction and polarized along the [110] direction, is small in comparison with those of the other shear and longitudinal modes. The Gruneisen parameter of this mode, and hence its vibrational anharmonicity, is much larger than those of the other long-wavelength acoustic phonon modes.