  We consider the low-temperature behavior of NMR relaxation rates $T_1$ and
$T_{2G}$ in the spin-1/2 antiferromagnetic chain. We find that $T_1 \propto
const$ and $T_{2G} \propto \sqrt{T}$, with logarithmic corrections. We
determine both constants and logarithmic terms by matching perturbative RG with
exact results, so that the final expressions for the relaxation rates do not
contain any free parameters. Our theoretical results are in excellent agreement
with NMR experiments in Sr$_2$CuO$_3$.
