Freestanding silicene has been predicted to possess graphene-like electronic structures with a Dirac cone; however, the gapless nature of silicene limits its applications in silicon-based electronics. On the other hand, it is also important to find a substrate that can support silicene, which keeps the required features thereof. Here, based on the first-principles calculations, we show that an alpha-phase bismuth-passivated Si surface, Le., Si(111) root 3 x root 3 - Bi, is semiconducting with a band-gap greater than 0.75 eV which could then be an appropriate surface to support silicene. For the adsorption of silicene on the passivated Si surface, silicene/Si(111) root 3 x root 3 - Bi, three adsorption sites of silicene, Le., the hollow site (H-site) and the two top sites (T-L-site and T-H-site), are discussed. Silicene absorbed on the H-site can open a band gap (0.174 eV) for the silicene layer, and the Dirac cone of silicene disappeared. For the T-L-site model, a semiconducting character of the system with a very small band gap (31 meV) is found. For the silicene on the TH-site, the interaction between the silicene layer and the passivated Si(111) surface is very weak, suggesting that such a Bi-passivated surface Si(111) root 3 x root 3 - Bi,can support silicene, meanwhile, keeping the main features of silicene thereof. The present study suggests that a Bi-passivated Si(111) surface could be used to support silicene and open a small band gap at the silicene layer.