  We report on the effects of hydrostatic pressure (HP) on the charge density
wave observed in underdoped cuprates. We studied YBa$_2$Cu$_3$O$_{6.6}$
($T_c$=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 Q$_{CDW}$ rapidly decrease as HP
is applied, resulting in the complete suppression of signatures of the CDW
below $\sim$1 GPa. Additional IXS measurements on YBa$_2$Cu$_3$O$_{6.75}$
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.
