  We report on a study of diffractive dijet production in $\bar{p}p$ collisions
at $\sqrt{s}=1.96$ TeV using the CDF II detector at the Fermilab Tevatron
$\bar{p}p$ collider. A data sample from 310 pb$^{-1}$ of integrated luminosity
collected by triggering on a high transverse energy jet, $E_T^{jet}$, in
coincidence with a recoil antiproton detected in a Roman pot spectrometer is
used to measure the ratio of single-diffractive to inclusive-dijet event rates
as a function of $x^{\bar p}$ of the interacting parton in the antiproton, the
Bjorken-$x$, $x^{\bar p}_{Bj}$, and a $Q^2\approx (E_T^{jet})^2$ in the ranges
$10^{-3}<x^{\bar p}_{Bj}<10^{-1}$ and $10^2<Q^2 <10^4$ GeV$^2$, respectively.
Results are presented for the region of $\bar p$-momentum-loss fraction
$0.03<\xi_{\bar p}<0.09$ and a four-momentum transfer squared $t_{\bar p}>-4$
GeV$^2$. The $t_{\bar p}$ dependence is measured as a function of $Q^2$ and
$x_{Bj}^{\bar p}$ and compared with that of inclusive single diffraction
dissociation. We find weak $x^{\bar p}_{Bj}$ and $Q^2$ dependencies in the
ratio of single diffractive to inclusive event rates, and no significant $Q^2$
dependence in the diffractive $t_{\bar p}$ distributions.
