We have studied the nonlinear conductivity \ensuremath{\sigma}(E,T) and the narrow-band noise in quasi-one-dimensional ${\mathrm{TaS}}_{3}$ at low temperatures down to liquid helium. We have found that the ratio between the charge-density-wave (CDW) current ${\mathit{I}}_{\mathit{c}}$ and the narrow-band-noise frequency f, ${\mathit{I}}_{\mathit{c}}$/f, begins to sharply decrease below \ensuremath{\sim}80 K with a slope correlated with the number of defects in the samples. We have carried out the analysis of our experimental data on the basis of the equation for phase slippage in narrow superconducting channels. We show that this equation well describes the \ensuremath{\sigma}(E,T) dependences at low temperatures in the high electric-field range above the threshold value. The value of the energy barrier for phase slippage has been estimated. We suggest that the decrease of the ${\mathit{I}}_{\mathit{c}}$/f ratio is mainly caused by the growth of the relaxation time of the CDW response at low temperatures. Our results confirm that the fundamental mechanism which determines the motion of the CDW and its excitations is the phase slippage both in the high- and in the low-temperature ranges.