Hadrons composed of more than three quarks, referred to as exotic states, play a key role in understanding the confinement mechanism of the strong interaction. Over the past two decades, several exotic hadrons have been experimentally observed, starting with the discovery of the $\chi_{c1}(3872)$ in 2003. In 2020, the LHCb collaboration observed the open-charm tetraquark states $T_{cs0}^{*}(2870)^0$ and $T_{cs1}^{*}(2900)^0$ in the $D^+K^-$ channel in the $B^- \rightarrow D^- D^+ K^-$ decay. This marked the first observation of exotic hadrons with a single charm quark, offering new insights into the quark binding mechanism. We report the observation of the $T_{cs0}^{*}(2870)^0$ state through a new decay channel $D^0 K_S^0$ in the $B^- \rightarrow D^- D^0 K_S^0$ decay, by performing an amplitude analysis. Using $9\, \mathrm{fb}^{-1}$ of proton-proton collision data collected by the \lhcb detector at center-of-mass energies of 7, 8, and 13\,TeV, we identified a resonant structure with spin-parity $0^+$ in the $D^0 K_S^0$ invariant mass spectrum with a significance of $5.3\sigma$. The state has a mass of $2883 \pm 11 \text{ MeV}c^2$ and a width of $87^{+22}_{-47} \pm 6$ MeV, which are consistent with the previously observed $T_{cs0}^{*}(2870)^0$ state. No evidence was found for the related $T_{cs1}^{*}(2900)^0$ state in this analysis. These findings contribute to a deeper understanding of open-charm exotic states.