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We measure the time-dependent $CP$ asymmetry parameters in ${B}^{0}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{K}^{0}$ based on a data sample of approximately $227\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B$-meson pairs recorded at the $\ensuremath{\Upsilon}(4S)$ resonance with the BABAR detector at the PEP-II $B$-meson Factory at SLAC. We reconstruct two-body ${B}^{0}$ decays to $\ensuremath{\phi}(1020){K}_{S}^{0}$ and $\ensuremath{\phi}(1020){K}_{L}^{0}$, and the three-body decay ${K}^{+}{K}^{\ensuremath{-}}{K}_{S}^{0}$ with …