  We have investigated the pressure dependence of the superconducting
transition temperature $T_{c}$ up to 18 GPa of
Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ single crystals ranging from the
highly underdoped through the nearly optimally doped to the highly overdoped
level. For all three samples studied, $T_{c}$ is found to increase initially
and then saturate at a critical pressure $P_{c}$ but decrease modestly with
further compression. Oxygen doping tends to reduce the increase in $T_{c}$ and
$P_{c}$. A new high-pressure phase diagram between the saturated $T_{c}$ and
$P_{c}$ is then obtained. Theoretical interpretation is given by using the
competition between the hole carrier density and pairing interaction strength
based on the high-pressure transport data of the resistivity and Hall
coefficient in this system.
