  We study the effect of Coulomb interactions on the conductance of a
single-mode quantum wire connecting two bulk leads. When the density of
electrons in the wire is very low, they arrange in a finite-length Wigner
crystal. In this regime the electron spins form an antiferromagnetic Heisenberg
chain with exponentially small coupling J. An electric current in the wire
perturbs the spin chain and gives rise to a temperature-dependent contribution
of the spin subsystem to the resistance. At low temperature T << J this effect
is small, and the conductance of the wire remains close to 2e^2/h. At T >> J
the spin effect reduces the conductance to e^2/h.
