There exists $C=C({\rm v},D)$ such that if $M^{4}$ satisfies $|{\rm Ric}_{M^{4}}|\leq 3$, ${\rm Vol}(B_{1}(p))>{\rm v}>0$ and ${\rm diam}(M^{n})\leq D$, then $M^{4}$ can have one of at most $C$ diffeomorphism types.