  The $B^{-}\to D^{+}K^{-}\pi^{-}$ decay is observed in a data sample
corresponding to $3.0~\rm{fb}^{-1}$ of $pp$ collision data recorded by the LHCb
experiment during 2011 and 2012. Its branching fraction is measured to be
${\cal B}(B^{-}\to D^{+}K^{-}\pi^{-}) = (7.31 \pm 0.19 \pm 0.22 \pm 0.39)
\times 10^{-5}$ where the uncertainties are statistical, systematic and from
the branching fraction of the normalisation channel $B^{-}\to
D^{+}\pi^{-}\pi^{-}$, respectively. An amplitude analysis of the resonant
structure of the $B^{-}\to D^{+}K^{-}\pi^{-}$ decay is used to measure the
contributions from quasi-two-body $B^{-}\to D_{0}^{*}(2400)^{0}K^{-}$,
$B^{-}\to D_{2}^{*}(2460)^{0}K^{-}$, and $B^{-}\to D_{J}^{*}(2760)^{0}K^{-}$
decays, as well as from nonresonant sources. The $D_{J}^{*}(2760)^{0}$
resonance is determined to have spin~1.
