Measurement of the Ratio of Branching Fractions $ {\cal B}(D^0 \to K^+
\pi^-)/{\cal B}(D^0 \to K^- \pi^+)$ and Prospects for Measuring $D^0 -
\bar{D}^0$ Mixing using the CDF II Detector

We present a measurement of $R_D$, the ratio of the branching fraction
for the doubly Cabibbo-suppressed decay $D^0 \to K^+ \pi^-$ to that
for the Cabibbo-favored decay $D^0 \to K^- \pi^+$.  A signal of $2005
\pm 104$ events for the decay $D^0 \to K^+ \pi^-$ is obtained using
the CDF~II detector at the Fermilab Tevatron collider. The data set
corresponds to an integrated luminosity of 0.35~fb$^{-1}$ produced in
$\bar{p}p$ collisions at $\sqrt{s}=1.96$~TeV.  Assuming no mixing and
no CP violation, we find $R_D = 4.05 \pm 0.21 (\rm stat) \pm 0.11
(\rm syst) \times 10^{-3}$.  This measurement is consistent with
the world average, and comparable in accuracy with the best
measurements from other experiments.
We discuss the prospects for measuring $D^0 -
\bar{D}^0$ mixing by comparing the time dependence of the doubly
Cabibbo-suppressed and Cabibbo-favored signals using a data set
corresponding to an integrated luminosity of 1 fb$^{-1}$.