  The Drude weight for the spin transport of the spin-1/2 $XXZ$ Heisenberg
chain in the critical regime is evaluated exactly for finite temperatures. We
combine the thermodynamic Bethe ansatz with the functional relations of type
$Y$-system satisfied by the row-to-row transfer matrices. This makes it
possible to evaluate the asymptotic behavior of the finite temperature spin
Drude weight with respect to the system size. As a result, the Drude weight
converges to the results obtained by Zotos (Phys. Rev. Lett. 82, 1764 (1999)),
however with very slow convergence upon increase of the system size. This
strong size dependence may explain that extrapolations from various numerical
approaches yield conflicting results.
