  In order to support the formal renormalization group arguments that the fixed
point action of an asymptotically free model gives cut--off independent
physical predictions in 1--loop perturbation theory, we calculate the finite
volume mass--gap $m(L)$ in the non--linear $\sigma$--model. No cut--off effect
of the type $g^4\left(a/L\right)^n$ is seen for any $n$. The results are
compared with those of the standard and tree level improved Symanzik actions.
