  We investigate dephasing in open quantum chaotic systems in the limit of
large system size to Fermi wavelength ratio, $L/\lambda_F >> 1$. We
semiclassically calculate the weak localization correction $g^{wl}$ to the
conductance for a quantum dot coupled to (i) an external closed dot and (ii) a
dephasing voltage probe. In addition to the universal algebraic suppression
$g^{wl} \propto (1+\tau_D/\tau_\phi)^{-1}$ with the dwell time $\tau_D$ through
the cavity and the dephasing rate $\tau_\phi^{-1}$, we find an exponential
suppression of weak localization by a factor $\propto
\exp[-\tilde{\tau}/\tau_\phi]$, with a system-dependent $\tilde{\tau}$. In the
dephasing probe model, $\tilde{\tau}$ coincides with the Ehrenfest time,
$\tilde{\tau} \propto \ln [L/\lambda_F]$, for both perfectly and partially
transparent dot-lead couplings. In contrast, when dephasing occurs due to the
coupling to an external dot, $\tilde{\tau} \propto \ln [L/\xi]$ depends on the
correlation length $\xi$ of the coupling potential instead of $\lambda_F$.
