Broadband optical waveguiding in an optical nanofiber can give great versatility to nanofiber-optic technology. To date, high-transmittance single-mode waveguiding in a single optical nanofiber in the full visible spectral range remains challenging. Here, by using a commercially available silica fiber with a single-mode cut-off wavelength smaller than 400 nm (Nufern, 405-HP) as the fiber preform, and taper-drawing the fiber into a nanofiber with a standard fabrication technique, we show that the nanofiber can support broadband single-mode waveguiding with high optical transmittance in the full visible spectral range. Quantitatively, a silica nanofiber with a uniform diameter between 210 and 270 nm can support single-mode optical waveguiding with transmittance higher than 96% (i.e., less than 0.18 dB in loss) from 400 to 800-nm wavelength. Such a broadband single-mode nanofiber may find applications including near-field optical coupling, optical interconnection, optical sensing, atom optics and fiber lasers in the broad visible spectral range.