We study electron transport along a strongly depleted one-dimensional channel and assume the limit of very weak disorder when electrons form the Wigner crystal. Pinning of this crystal by a weak potential of single impurity is considered in the quantum case. Zero-point oscillations of the crystal significantly diminish the pinning barrier. However, it remains finite at moderate densities of electrons (naB 0.5), in spite of the absence of the long-range order in one dimension. Charge transfer in the system occurs due to thermally assisted tunneling of the crystal through the pinning barrier, which results in a power-law temperature dependence of the conductance (T). The nonlinear conductance (U) at T=0 obeys a similar law.