  We use matrix model results to investigate the Sine-Gordon model coupled to
two dimensional gravity. For relevant (in the RG sense) potentials, we show
that the $c=1$ string, which appears in the ultraviolet limit of this model,
flows to a set of decoupled $c=0$ (pure gravity) models in the infrared. The
torus partition sum, which was argued previously to count the number of string
degrees of freedom and hence satisfy a new $c$ -- theorem, is shown to be a
monotonically decreasing function of the scale (given by the quantum area of
the world-sheet).
