Search for D0D0bar Mixing and a Measurement of the Doubly Cabibbo-suppressed Decay Rate in D0 -> K pi Decays

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