  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.
