An amplitude analysis of B^{-}→D^{-}D^{0}K_{S}^{0} decays is performed using proton-proton collision data, corresponding to an integrated luminosity of 9 fb^{-1}, collected with the LHCb detector at center-of-mass energies of 7, 8, and 13 TeV. A resonant structure of spin-parity 0^{+} is observed in the D^{0}K_{S}^{0} invariant-mass spectrum with a significance of 5.3σ. The mass and width of the state, modeled with a Breit-Wigner line shape, are determined to be 2883±11±8 MeV/c^{2} and 87_{-47}^{+22}±17 MeV, respectively, where the first uncertainties are statistical and the second systematic. These properties and the quark content are consistent with those of the open-charm tetraquark candidate T_{cs0}^{*}(2870)^{0} observed previously in the D^{+}K^{-} final state of the B^{-}→D^{-}D^{+}K^{-} decay. This result confirms the existence of the T_{cs0}^{*}(2870)^{0} state in a new decay mode. The T_{cs1}^{*}(2900)^{0} state, reported in the B^{-}→D^{-}D^{+}K^{-} decay, is also searched for in the D^{0}K_{S}^{0} invariant-mass spectrum of the B^{-}→D^{-}D^{0}K_{S}^{0} decay, without finding evidence for it.