To test the existence of a possible radial gradient in oxygen abundances within the Local Group dwarf irregular galaxy NGC 6822, we have obtained optical spectra of 19 nebulae with the EFOSC2 spectrograph on the 3.6-m telescope at ESO La Silla. The extent of the measured nebulae spans galactocentric radii in the range between 0.05 kpc and 2 kpc (over four exponential scale lengths). In five H II regions (Hubble I, Hubble V, Kalpha, Kbeta, KD28e), the temperature-sensitive [O III] 4363 emission line was detected, and direct oxygen abundances were derived. Oxygen abundances for the remaining H II regions were derived using bright-line methods. The oxygen abundances for three A-type supergiant stars are slightly higher than nebular values at comparable radii. Linear least-square fits to various subsets of abundance data were obtained. When all of the measured nebulae are included, no clear signature is found for an abundance gradient. A fit to only newly observed H II regions with [O III] 4363 detections yields an oxygen abundance gradient of -0.14 +/- 0.07 dex/kpc. The gradient becomes slightly more significant (-0.16 +/- 0.05 dex/kpc) when three additional H II regions with [O III] 4363 measurements from the literature are added. Assuming no abundance gradient, we derive a mean nebular oxygen abundance 12+log(O/H) = 8.11 +/- 0.10 from [O III] 4363 detections in the five H II regions from our present data; this mean value corresponds to [O/H] = -0.55.
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