Topological superconductors have long been predicted to host Majorana zero modes which obey non-Abelian statistics and have potential for realizing non-decoherence topological quantum computation. However, material realization of topological superconductors is still a challenge in condensed matter physics. Utilizing high-resolution angle-resolved photoemission spectroscopy and first-principles calculations, we predict and then unveil the coexistence of topological Dirac semimetal and topological insulator states in the vicinity of Fermi energy ($E_F$) in the titanium-based oxypnictide superconductor BaTi$_2$Sb$_2$O. Further spin-resolved measurements confirm its spin-helical surface states around $E_F$, which are topologically protected and give an opportunity for realization of Majorana zero modes and Majorana flat bands in one material. Hosting dual topological superconducting states, the intrinsic superconductor BaTi$_2$Sb$_2$O is expected to be a promising platform for further investigation of topological superconductivity.
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