  We derive a phase diagram for the pseudogap onset temperature $T^*$
(associated with the breakdown of the Fermi liquid state, due to strong pairing
correlations) and the superconducting instability, $T_c$, as a function of
variable pairing strength. Our diagrammatic approach to the BCS - Bose-Einstein
cross-over problem self consistently treats the coupling between the single
particle and pair propagators, and leads to a continuous evolution of these
propagators into the standard $T<T_c$ counterparts. A rich structure is found
in $T_c$ which reflects the way in which the superconducting instability at
$T_c$ is affected by the pseudogap $\Delta_{pg}$. An important consequence of
Cooper-pair- induced pseudogaps is that the magnitude of $T_c$ is sustained,
even when $\Delta_{pg}>T_c$.
