Majorana bound states are zero-energy excitations predicted to localize at the edge of a topological superconductor, a state of matter that can form when a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin-orbit interaction. With the goal of realizing a one-dimensional topological superconductor, we have fabricated ferromagnetic iron atomic chains on the surface of superconducting lead . Using high-resolution spectroscopic imaging techniques, we show that the onset of superconductivity, which partly gaps the electronic density of states in the bulk of the chains, is accompanied by the appearance of zero-energy end-states. This spatially resolved signature provides evidence, corroborated by other observations and theoretical modeling , for the formation of a topological phase and edge-bound Majorana states in this system. Our results demonstrates that atomic chains are viable platform for future experiments to manipulate Majorana bound states  and to realize other 1D and 2D topological superconducting phases.