  Form Factor Perturbation Theory is applied to study the spectrum of the O(3)
non--linear sigma model with the topological term in the vicinity of $\theta =
\pi$. Its effective action near this value is given by the non--integrable
double Sine--Gordon model. Using previous results by Affleck and the explicit
expressions of the Form Factors of the exponential operators $e^{\pm
i\sqrt{8\pi} \phi(x)}$, we show that the spectrum consists of a stable triplet
of massive particles for all values of $\theta$ and a singlet state of higher
mass. The singlet is a stable particle only in an interval of values of
$\theta$ close to $\pi$ whereas it becomes a resonance below a critical value
$\theta_c$.
