  We present a study of the Drude weight $D(T)$ of the spin-1/2 $XXZ$ chain in
the gapless regime. The thermodynamic Bethe ansatz (TBA) is applied in two
different ways. In the first application we employ the particle basis of
magnons and their bound states. In this case we rederive and considerably
extend earlier work in the literature. However, in the course of our
investigation we find arguments that cast doubt on the applicability of the TBA
in this case. In a second application by use of the spinon and anti-spinon
particle basis we obtain completely different results. Only for anisotropy
parameter $\Delta$ close to 0 we find that $D(T)$ is a monotonously decaying
function of temperature. For $\Delta$ close to 1 the behaviour is entirely
different showing a finite temperature maximum. Also for the isotropic
antiferromagnetic chain ($\Delta=1$) the results for $D(T)$ are finite for T=0
as well as for $T>0$ with an infinite positive slope at T=0.
