  In a previous paper, we reported on the discovery of more than a hundred new
Cepheid variables in the Sculptor Group spiral NGC 300 from wide-field images
taken in the B and V photometric bands at ESO/La Silla. In this paper, we
present additional VI data, derive improved periods and mean magnitudes for the
variables, and construct period-luminosity relations in the V, I and the
reddening-independent (V-I) Wesenheit bands using 58 Cepheid variables with
periods between 11 and 90 days. We obtain tightly defined relations, and by
fitting the slopes determined for the LMC Cepheids by the OGLE II Project we
obtain reddening-corrected distances to the galaxy in all bands. We adopt as
our best value the distance derived from the reddening-free Wesenheit
magnitudes, which is 26.43 $\pm$ 0.04 (random) $\pm$ 0.05 (systematic) mag. We
argue that our current distance result for NGC 300 is the most accurate which
has so far been obtained using Cepheid variables, and that it is largely free
from systematic effects due to metallicity, blending, and sample selection. It
agrees very well with the recent distance determination from the tip of the red
giant branch method obtained from HST data by Butler et al. (2004), and it is
consistent with the Cepheid distance to NGC 300 which was derived by Freedman
et al. (2001) from CCD photometry of a smaller sample of stars.
