  We present a measurement of relative partial widths and decay rate CP
asymmetries in $K^-K^+$ and $\pi^-\pi^+$ decays of $\Do$ mesons produced in
$p\bar{p}$ collisions at $\sqrt{s} = 1.96 \TeV$. We use a sample of
$2\times10^5$ $D^{\ast+}\myto D^0\pi^+$ (and charge conjugate) decays with the
$D^0$ decaying to $K^-\pi^+$, $K^-K^+$, and $\pi^-\pi^+$, corresponding to $123
{\rm pb}^{-1}$ of data collected by the Collider Detector at Fermilab II
experiment at the Fermilab Tevatron collider. No significant direct CP
violation is observed. We measure $\KKtokpi = 0.0992 \pm 0.0011 \pm 0.0012$,
$\PIpitokpi = 0.03594 \pm 0.00054 \pm 0.00040$, $A_{CP}(K^-K^+) = (2.0 \pm 1.2
\pm 0.6) %$, and $A_{CP}(\pi^-\pi^+) = (1.0 \pm 1.3 \pm 0.6) %$, where, in all
cases, the first uncertainty is statistical and the second is systematic.
