  We consider the isotropic $S=1$ Heisenberg chain with a finite Haldane gap
$\Delta$ and use state-of-the-art numerical techniques to investigate its
dynamical properties at finite temperature, focusing on the nuclear
spin-lattice relaxation rate $1/T_1$ measured in nuclear magnetic resonance
(NMR) experiments for instance. In particular, we analyze the contributions
from modes with momenta close to $q\approx 0$ and $q\approx \pi$ as a function
of temperature. At high-temperature, we observe spin diffusion with a
non-trivial exponent. At low-temperature, we argue that a simple activated
behavior $1/T_1 \propto\exp(-\Delta/T)$ can only be observed at temperatures
much smaller than the gap $\Delta$.
