  Measurements of the spin-lattice relaxation rate 1/T_1 by nuclear magnetic
resonance for the one-dimensional Heisenberg antiferromagnet Sr_2CuO_3 have
provided evidence for a diffusion-like contribution at finite temperature and
small wave-vector. By analyzing real-time data for the auto- and
nearest-neighbor spin-spin correlation functions obtained by the density-matrix
renormalization group I show that such a contribution indeed exists for
temperatures T>J, where J is the coupling constant, but that it becomes
exponentially suppressed for T << J. I present evidence that the
frequency-dependence of 1/T_1 in the Heisenberg case is smoothly connected to
that in the free fermion case where the exponential suppression of the
diffusion-like contribution is easily understood.
