  (Shortened version) We report optical spectroscopy of 16 H II regions in NGC
1705 and [O III] 4363 detections for the first time in five H II regions. The
resulting mean oxygen abundance derived directly from measured electron
temperatures is 12+log(O/H) = 8.21 +/- 0.05, which corresponds to [O/H] =
-0.45, or 35% of the solar value. There are no significant spatial
inhomogeneities in [O III] 4363 oxygen abundances from H II regions at a radius
approximately 10 arcsec from the super star cluster. In H II regions where [O
III] 4363 was not measured, oxygen abundances derived with bright-line methods
(accurate only to 0.2 dex) are in agreement with direct values of the oxygen
abundance. Faint narrow He II 4686 emission is found in two H II regions, but
the implied contribution from O^+3 to the total oxygen abundance is only 0.01
dex. Interestingly, the nitrogen-to-oxygen abundance ratio in the ionized H II
gas agrees with the value for the neutral H I, even though the metallicity of
the neutral gas may be a factor of six lower than that of the ionized gas. This
may be indicative of low-metallicity gas in the halo of the galaxy. Extinction
values, A_V, derived from observed Balmer line ratios along lines of sight to H
II regions are in the range between zero and 0.9 mag. With respect to the
metallicity-luminosity and metallicity-gas fraction diagnostics, the measured
oxygen abundance for NGC 1705 is comparable to Local Group dwarf irregulars at
a given luminosity and gas fraction.
