  We calculate the finite volume mass gap $M(L)$ at 3-loop level in the
non-linear O($n$) $\sigma$-model in two dimensions in small volumes. By
applying the Monte Carlo measurements of the running coupling $\bar
g^2(L)=2nM(L)L/(n-1)$ by L\"uscher, Weisz and Wolff measured in units of the
physical mass gap $m$, the result is used to determine $m$ in units of the
$\Lambda$-parameter in the O(3) and O(4) models. Our determinations show good
agreement with those by Hasenfratz, Maggiore and Niedermayer in both models. We
note that this manuscript has been revised in our paper hep-lat/9810025 by
using the corrected four-loop $\beta$-function on the lattice.
