  The spin-$1/2$ XXZ chain is an integrable lattice model and parts of its spin
current can be protected by local conservation laws for anisotropies
$-1<\Delta<1$. In this case, the Drude weight $D(T)$ is non-zero at finite
temperatures $T$. Here we obtain analytical results for $D(T)$ at low
temperatures for zero external magnetic field and anisotropies
$\Delta=\cos(n\pi/m)$ with $n,m$ coprime integers, using the thermodynamic
Bethe ansatz. We show that to leading orders
$D(T)=D(0)-a(\Delta)T^{2K-2}-b_1(\Delta)T^2$ where $K$ is the Luttinger
parameter and the prefactor $a(\Delta)$, obtained in closed form, has a fractal
structure as function of anisotropy $\Delta$. The prefactor $b_1(\Delta)$, on
the other hand, does not have a fractal structure and can be obtained in a
standard field-theoretical approach. Including both temperature corrections, we
obtain an analytic result for the low-temperature asymptotics of the Drude
weight in the entire regime $-1<\Delta=\cos(n\pi/m)<1$.
