  The spin-lattice relaxation rate $1/T_1$ and the spin echo decay rate
$1/T_{2G}$ for the spin-$1\over 2$ antiferromagnetic Heisenberg chain are
calculated using quantum Monte Carlo and maximum entropy analytic continuation.
The results are compared with recent analytical calculations by Sachdev. If the
nuclear hyperfine form factor $A_q$ is strongly peaked around $q=\pi$ the
predicted low-temperature behavior [$1/T_1 \sim \ln{^{1/2}(1/T)}$, $1/T_{2G}
\sim \ln{^{1/2}(1/T)}/\sqrt{T}$] extends up to temperatures as high as $T/J
\approx 0.5$. If $A_q$ has significant weight for $q \approx 0$ there are large
contributions from diffusive long-wavelength processes not taken into account
in the theory, and very low temperatures are needed in order to observe the
asymptotic $T \to 0$ forms.
