To realize a one-dimensional (1D) system with strong spin-orbit coupling is a big challenge in modern physics, since the electrons in such a system are predicted to exhibit exotic properties unexpected from the 2D or 3D counterparts, while it was difficult to realize genuine physical properties inherent to the 1D system. We demonstrate the first experimental result that directly determines the purely 1D band structure by performing spin-resolved angle-resolved photoemission spectroscopy of Bi islands on a silicon surface that contains a metallic 1D edge structure with unexpectedly large Rashba-type spin-orbit coupling suggestive of the nontopological nature. We have also found a sizable out-of-plane spin polarization of the 1D edge state, consistent with our first-principles band calculations. Our result provides a new platform to realize exotic quantum phenomena at the 1D edge of the strong spin-orbit-coupling systems.