  We report on the observation of $D^0$--$\bar{D}^0$ oscillations by measuring
the time-dependent ratio of yields for the rare decay $D^0 \rightarrow
K^+\pi^-$ to the favored decay $D^0 \rightarrow K^-\pi^+$ at the Collider
Detector at Fermilab (CDF). Using 9.6 fb$^{-1}$ of integrated luminosity of
$\sqrt{s}$ = 1.96 TeV $p$$\bar{p}$ collisions recorded in the full CDF Run II,
the signals of $7.6 \times 10^6$ $D^0\rightarrow K^-\pi^+$ and $33 \times 10^3$
$D^0\rightarrow K^+\pi^-$ decays are reconstructed in $D^{*}$-tagged events,
with proper decay times between 0.75 and 10 mean $D^0$ lifetimes. We measure
the mixing parameters $x'^2 = (0.08 \pm 0.18)\times 10^{-3}$, $y' = (4.3 \pm
4.3) \times 10^{-3}$, and $R_D = (3.51 \pm 0.35) \times 10^{-3}$. Our results
are consistent with standard model expectations and similar results from
proton-proton collisions and exclude the no-mixing hypothesis with a
significance equivalent to 6.1 standard deviations.
