  We present results of a search for $D^0$-$\kern 0.2em\bar{\kern -0.2em
D}{}^0$~mixing and a measurement of $R_{\rm D}$, the ratio of doubly
Cabibbo-suppressed decays to Cabibbo-favored decays, based on an analysis of
$D^0 \to K^{+}\pi^{-}$ decays in 57.1~${fb}^{-1}$ of data collected at or just
below the $\Upsilon{(4S)}$ resonance with the {\slshape BABAR} detector at the
PEP-II collider. Our results are compatible with no mixing and no $CP$
violation. At the 95% C.L., allowing for CP violation, we find the mixing
parameters ${x^{\prime}}^2<0.0022$ and $-0.056<y^{\prime}<0.039$, and the
mixing rate $R_{\rm M} < 0.16%$. In the limit of no mixing, $R_{\rm D} = (0.357
\pm 0.022 \hbox{(stat.)} \pm 0.027 \hbox{(syst.)})%$ and the $CP$~violating
asymmetry \mbox{$A_{\rm D} = 0.095 \pm 0.061 \hbox{(stat.)} \pm 0.083
\hbox{(syst.)}$.}
