  We consider a dynamical system described by the differential equation
$\dot{Y}_t=-U'(Y_t)$ with a unique stable point at the origin. We perturb the
system by the L\'evy noise of intensity $\varepsilon$ to obtain the stochastic
differential equation $dX^{\varepsilon}_t=-U'(X^{\varepsilon}_{t-})
dt+\varepsilon dL_t.$ The process $L$ is a symmetric L\'evy process whose jump
measure $\nu$ has exponentially light tails,
$\nu([u,\infty))\sim\exp(-u^{\alpha})$, $\alpha>0$, $u\to \infty$. We study the
first exit problem for the trajectories of the solutions of the stochastic
differential equation from the interval $(-1,1)$. In the small noise limit
$\varepsilon\to0$, the law of the first exit time $\sigma_x$, $x\in(-1,1)$, has
exponential tail and the mean value exhibiting an intriguing phase transition
at the critical index $\alpha=1$, namely,
$\ln\mathbf{E}\sigma\sim\varepsilon^{-\alpha}$ for $0<\alpha<1$, whereas
$\ln\mathbf{E}\sigma\sim\varepsilon^{- 1}|\ln\varepsilon|^{1-{1}/{\alpha}}$ for
$\alpha>1$.
