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We performed angle-resolved photoelectron spectroscopy of the Bi(111) surface to demonstrate that this surface supports edge states of non-trivial topology. Along the $\bar{\Gamma }\skew3\bar{M}$ -direction of the surface Brillouin zone, a surface-state band disperses from the projected bulk valence bands at $\bar{\Gamma }$ to the conduction bands at $\skew3\bar{M}$ continuously, indicating the non-trivial topological order of three-dimensional Bi bands. We ascribe this finding to the absence of band inversion at the L point of the bulk Bi Brillouin zone. According to our analysis, a modification of tight-binding parameters can account for the non-trivial band structure of Bi without any other significant change in other physical properties.