We report on the effects of hydrostatic pressure (HP) on the charge density wave observed in underdoped cuprates. We studied YBa2Cu3O6.6 (Tc=61 K) using high-resolution inelastic x-ray scattering (IXS), and reveal an extreme sensitivity of the phonon anomalies related to the charge density wave (CDW) order to HP. The amplitudes of the normal state broadening and superconductivity induced phonon softening at QCDW rapidly decrease as HP is applied, resulting in the complete suppression of signatures of the CDW below ∼1 GPa. Additional IXS measurements on YBa2Cu3O6.75 demonstrate that this very rapid effect cannot be explained by pressure-induced modification of the doping level and highlight the different role of external pressure and doping in tuning the phase diagram of the cuprates. Our results provide new insights into the mechanisms underlying the CDW formation and its interplay with superconductivity.The complex phase diagrams of correlated electron systems are shaped by the interplay between almost degenerate electronic states. In the high-T c superconducting cuprates, charge density wave (CDW) modulations have been ubiquitously observed, appearing as a generic feature of the moderately doped CuO 2 plane [1-3, 5-10]. While the doping and magnetic field dependence of the CDW order and its competition with superconductivity have been thoroughly studied [2-4, 11, 12], the actual impact of CDW on the superconducting properties remains unclear. Recent studies of stripe ordered La 1.875 Ba 0.125 CuO 4 showed that the application of pressure increases T c , while drastically suppressing static spin and charge orders, as well as the local tilts of the CuO 6 octahedra which act as stripe pinning centers [13,14]. However, no direct measurements of the effect of pressure on the CDW in the other families of cuprates have been reported to date.High pressures have been extensively used as a tuning parameter of the superconducting properties of the cuprates, and yielded the highest T c ever reported in these materials [15,16] (and that held the world record until very recently, when it got surpassed by highly pressurized H 3 S [17]). In YBa 2 Cu 3 O 6+x , the few available structural studies revealed that hydrostatic pressure (HP) brings the superconducting CuO 2 planes and the Cu-O chains closer together, leading to a net increase of the number of holes per Cu atom in the CuO 2 planes [18]. Recently it was shown that HP up to ∼ 30 GPa applied on fully oxygenated YBa 2 Cu 3 O 7 , provides access to the so-far unreachable highly overdoped and non-superconducting phase of this material [19]. On the other hand, in the underdoped region of the phase diagram and particularly in the region of the T c vs doping plateau where the CDW is best developed, a large increase of T c is observed under HP, yielding T c largely exceeding that of the optimally doped compound. The effective HP dependence of T c reflects the combination of pressure-induced doping with other HP effects, on e.g. superexchange interactions, or oxygen ordering [20...