  An effective, low temperature, classical model for spin transport in the
one-dimensional, gapped, quantum $O(3)$ non-linear $\sigma$-model is developed.
Its correlators are obtained by a mapping to a model solved earlier by Jepsen.
We obtain universal functions for the ballistic-to-diffusive crossover and the
value of the spin diffusion constant, and these are claimed to be exact at low
temperatures. Implications for experiments on one-dimensional insulators with a
spin gap are noted.
