  We study the finite-temperature dynamical spin susceptibility of the
one-dimensional (generalized) anisotropic Heisenberg model within the
hydrodynamic regime of small wave vectors and frequencies. Numerical results
are analyzed using the memory function formalism with the central quantity
being the spin-current decay rate gamma(q,omega). It is shown that in a generic
nonintegrable model the decay rate is finite in the hydrodynamic limit,
consistent with normal spin diffusion modes. On the other hand, in the gapless
integrable model within the XY regime of anisotropy Delta < 1 the behavior is
anomalous with vanishing gamma(q,omega=0) proportional to |q|, in agreement
with dissipationless uniform transport. Furthermore, in the integrable system
the finite-temperature q = 0 dynamical conductivity sigma(q=0,omega) reveals
besides the dissipationless component a regular part with vanishing
sigma_{reg}(q=0,omega to 0) to 0.
