  A formula for the mass-gap of the supersymmetric O($N$) sigma model ($N>4$)
in two dimensions is derived: $m/\Lambda_{\overline{\rm
MS}}=2^{2\Delta}\sin(\pi\Delta)/(\pi\Delta)$, where $\Delta=1/(N-2)$ and $m$ is
the mass of the fundamental vector particle in the theory. This result is
obtained by comparing two expressions for the free-energy density in the
presence of a coupling to a conserved charge; one expression is computed from
the exact S-matrix of Shankar and Witten via the the thermodynamic Bethe ansatz
and the other is computed using conventional perturbation theory. These
calculations provide a stringent test of the S-matrix, showing that it
correctly reproduces the universal part of the beta-function and resolving the
problem of CDD ambiguities.
