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.
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