  We extend the sphere theorem of \cite{CGY03} to give a conformally invariant
characterization of $(\mathbb{CP}^2, g_{FS})$. In particular, we introduce a
conformal invariant $\beta(M^4,[g]) \geq 0$ defined on conformal four-manifolds
satisfying a `positivity' condition; it follows from \cite{CGY03} that if $0
\leq \beta(M^4,[g]) < 4$, then $M^4$ is diffeomorphic to $S^4$. Our main result
of this paper is a `gap' result showing that if $b_2^{+}(M^4) > 0$ and $4 \leq
\beta(M^4,[g]) < 4(1 + \epsilon)$ for $\epsilon > 0$ small enough, then $M^4$
is diffeomorphic to $\mathbb{CP}^2$. The Ricci flow is used in a crucial way to
pass from the bounds on $\beta$ to pointwise curvature information.
