  We report transport measurements under very high current densities $j$, up to
$\sim10^8$~A/cm$^2$, of quasi-one-dimensional charge-density wave (CDW)
conductors NbSe$_3$ and TaS$_3$. Joule heating has been minimized by using a
point-contact configuration or by measuring samples with extremely small
cross-sections. Above $j_c \approx 10^7$~A/cm$^2$ we find evidence for
suppression of the Peierls gap and development of the metallic state. The
critical CDW velocity corresponding with $j_0$ is comparable with the sound
velocity, and with $\Delta/ \hbar k_F$ ($k_F$ is the Fermi wave vector), which
corresponds to the depairing current. Possible scenarios of the Peierls state
destruction are discussed.
