  The denaturation transition which takes place in circular DNA is analyzed by
extending the Poland-Scheraga model to include the winding degrees of freedom.
We consider the case of a homopolymer whereby the winding number of the double
stranded helix, released by a loop denaturation, is absorbed by
\emph{supercoils}. We find that as in the case of linear DNA, the order of the
transition is determined by the loop exponent $c$. However the first order
transition displayed by the PS model for $c>2$ in linear DNA is replaced by a
continuous transition with arbitrarily high order as $c$ approaches 2, while
the second-order transition found in the linear case in the regime $1<c\le2$
disappears. In addition, our analysis reveals that melting under fixed linking
number is a \emph{condensation transition}, where the condensate is a
macroscopic loop which appears above the critical temperature.
