The doubly charmed exotic state $T_{cc}$ recently discovered by the LHCb Collaboration could well be a $DD^{*}$ molecular state long predicted in various theoretical models, in particular, the $DD^*$ isoscalar axial vector molecular state predicted in the one-boson-exchange model. In this work, we study the $DDD^*$ system in the Gaussian Expansion Method with the $DD^*$ interaction derived from the one-boson-exchange model and constrained by the precise binding energy of $273pm63$ keV of $T_{cc}$ with respect to the $D^{*+}D^0$ threshold. We show the existence of a $DDD^*$ state with a binding energy of a few hundred keV and spin-parity $1^-$. Its main decay modes are $DDDpi$ and $DDDgamma$. The existence of such a state could in principle be confirmed with the upcoming LHC data and will unambiguously determine the nature of the $T_{cc}^+$ state and of the many exotic state of similar kind, thus deepening our understanding of the non-perturbative strong interaction.
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