A convex projective surface is the quotient of a properly convex open $\O$ of $\P$ by a discret subgroup $\G$ of $\s$. We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann's measure. We deduce of this that if $\O$ is not a triangle then $\O$ is strictly convex, with $\Cc^1$ boundary and that a convex projective surface $S$ is of finite volume if and only if the dual surface is of finite volume.