  We report a study of the suppressed decay $B^{-} \to
[K^{+}\pi^{-}]_{D}K^{-}$(and its charge-conjugate mode) at Belle, where
$[K^{+}\pi^{-}]_{D}$ indicates that the $K^{+}\pi^{-}$ pair originates from a
neutral $D$ meson. A data sample containing 274 million $B\bar{B}$ pairs
recorded at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB
asymmetric $e^{+}e^{-}$ storage ring is used. This decay mode can be used to
extract the CKM angle $\phi_{3}$ using the so-called Atwood-Dunietz-Soni
method. The signal for $B^{-} \to [K^{+}\pi^{-}]_{D}K^{-}$ has $2.7\sigma$
statistical significance, and we set a limit on the ratio of B decay amplitudes
$r_B < 0.28$ at the 90% confidence level. We observe a signal with $5.8\sigma$
statistical significance in the related mode, $B^{-} \to
[K^{+}\pi^{-}]_{D}\pi^{-}$.
