  (Abridged) We have obtained new spectrophotometric data for 28 HII regions in
the spiral galaxy NGC 300, a member of the nearby Sculptor Group. The detection
of auroral lines, including [OIII]4363, [SIII]6312 and [NII]5755, has allowed
us to measure electron temperatures and direct chemical abundances for the
whole sample. We determine for the first time in this galaxy a radial gas-phase
oxygen abundance gradient based solely on auroral lines, and obtain the
following least-square solution: 12+log(O/H)=8.57-0.41 R/R25, where the
galactocentric distance is expressed in terms of the isophotal radius R25. The
gradient corresponds to -0.077 dex/kpc, and agrees very well with the
galactocentric trend in metallicity obtained for 29 B and A supergiants in the
same galaxy. The intercept of the regression for the nebular data virtually
coincides with the intercept obtained from the stellar data. This allows little
room for depletion of nebular oxygen onto dust grains, although in this kind of
comparison we are somewhat limited by systematic uncertainties, such as those
related to the atomic parameters used to derive the chemical compositions.
  We discuss the implications of our result with regard to strong-line
abundance indicators commonly used to estimate the chemical compositions of
star-forming galaxies, such as R23. By applying a few popular calibrations of
these indices based on grids of photoionization models on the NGC 300 HII
region fluxes we find metallicities that are higher by 0.3 dex (a factor of
two) or more relative to our nebular (Te-based) and stellar ones.
  We confirm a metallicity dependence of the `softness' parameter
eta=(O+/O++)/(S+/S++), in the sense that softer stellar continua are found at
high metallicity.
