Topological superconductors are predicted to host exotic Majorana states that obey non-Abelian statistics and can be used to implement a topological quantum computer. Most of the proposed topological superconductors are realized in difficult-to-fabricate heterostructures at very low temperatures. By using high-resolution spin-resolved and angle-resolved photoelectron spectroscopy, we find that the iron-based superconductor FeTe1-xSex (x = 0.45; superconducting transition temperature T-c = 14.5 kelvin) hosts Dirac-cone-type spin-helical surface states at the Fermi level; the surface states exhibit an s-wave superconducting gap below T-c. Our study shows that the surface states of FeTe0.55Se0.45 are topologically superconducting, providing a simple and possibly high-temperature platform for realizing Majorana states.