  We study the exit-time from a domain of a self-interacting diffusion, where
the Brownian motion is replaced by $\sigma B_t$ for a constant $\sigma$. The
first part of this work consists in showing that the rate of convergence (of
the occupation measure of the self-interacting process toward some explicit
Gibbs measure) previously obtained in \cite{kk-ejp} for a convex confinment
potential $V$ and a convex interaction potential can be bounded uniformly with
respect to $\sigma$. Then, we prove an Arrhenius-type law for the first
exit-time from a domain (satisfying classical hypotheses of Freidlin-Wentzell
theory).
