  An $S$-matrix is proposed for the two dimensional O(3) $\sigma$-model with a
dynamical $\theta$-term (axion model). Exploiting an Abelian T-duality
transformation connecting the axion model to an integrable SU(2)$\times$U(1)
symmetric principal $\sigma$-model, strong evidence is presented for the
correctness of the proposed $S$-matrix by comparing the perturbatively
calculated free energies with the ones based on the Thermodynamical Bethe
Ansatz. This T-duality transformation also leads to a new Lax-pair for both
models. The quantum non-integrability of the O(3) $\sigma$-model with a {\sl
constant} $\theta$-term, in contradistinction to the axion model, is
illustrated by calculating the $2\to3$ particle production amplitude to lowest
order in $\theta$.
