We study the clustering of galaxies detected at i < 22.5 in the Science Verification observations of the Dark Energy Survey (DES). Two-point correlation functions are measured using 2.3 × 106 galaxies over a contiguous 116 deg2 region in five bins of photometric redshift width Δz = 0.2 in the range 0.2 < z < 1.2. The impact of photometric redshift errors is assessed by comparing results using a template-based photo-z algorithm (BPZ) to a machine-learning algorithm (TPZ). A companion paper presents maps of several observational variables (e.g. seeing, sky brightness) which could modulate the galaxy density. Here we characterize and mitigate systematic errors on the measured clustering which arise from these observational variables, in addition to others such as Galactic dust and stellar contamination. After correcting for systematic effects, we measure galaxy bias over a broad range of linear scales relative to mass clustering predicted from the Planck Λ cold dark matter model, finding agreement with the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) measurements with χ2 of 4.0 (8.7) with 5 degrees of freedom for the TPZ (BPZ) redshifts. We test a ‘linear bias’ model, in which the galaxy clustering is a fixed multiple of the predicted non-linear dark matter clustering. The precision of the data allows us to determine that the linear bias model describes the observed galaxy clustering to 2.5 per cent accuracy down to scales at least 4–10 times smaller than those on which linear theory is expected to be sufficient.