Effects of K-\bar K mixing on determining gamma from B^\pm -> DK^\pm

The decay $B^{\pm}\to DK^{\pm}$ followed by the subsequent decay of the $D$ meson into final states involving a neutral kaon can be used to determine the CKM angle $\gamma$. We study CP violation effects due to mixing and decay of the final state kaon. We find that ignoring these effects produce a shift in $\gamma$ of order $\epsilon_{K}/r_{B}$, an enhancement of $1/r_B$ compared to the naive expectation. We then show how to take these effects into account such that, in principle, they will not introduce any theoretical error in the extraction of $\gamma$.