The specific heat, C, of a single crystal of CeCu6 has been measured between 0.06 and 20K to pressures, P, of 8.8 kbar, and between 0.3 and 20K in magnetic fields, H, to 7.5T. At zero H and P, a minimum in C/T is observed near 8K. Below that temperature C/T increases, and below 1K is linear in T, C/T=1.67-0.67T J/mole K2. No maximum in C/T is found. This increase in C/T is reduced by a factor of three at 8.8 kbar and virtually completely suppressed for H = 7.5T along the [001] axis. With H ∥ [100] and [010] axes the effect on C is small.