The Gaia-ESO Survey is a large public spectroscopic survey that aims to derive radial velocities and fundamental parameters of about 10^5 Milky Way stars in the field and in clusters. Observations are carried out with the multi-object optical spectrograph FLAMES, using simultaneously the medium resolution (R~20,000) GIRAFFE spectrograph and the high resolution (R~47,000) UVES spectrograph. In this paper, we describe the methods and the software used for the data reduction, the derivation of the radial velocities, and the quality control of the FLAMES-UVES spectra. Data reduction has been performed using a workflow specifically developed for this project. This workflow runs the ESO public pipeline optimizing the data reduction for the Gaia-ESO Survey, performs automatically sky subtraction, barycentric correction and normalisation, and calculates radial velocities and a first guess of the rotational velocities. The quality control is performed using the output parameters from the ESO pipeline, by a visual inspection of the spectra and by the analysis of the signal-to-noise ratio of the spectra. Using the observations of the first 18 months, specifically targets observed multiple times at different epochs, stars observed with both GIRAFFE and UVES, and observations of radial velocity standards, we estimated the precision and the accuracy of the radial velocities. The statistical error on the radial velocities is sigma~0.4 km s^-1 and is mainly due to uncertainties in the zero point of the wavelength calibration. However, we found a systematic bias with respect to the GIRAFFE spectra (~0.9 km s^-1) and to the radial velocities of the standard stars (~0.5 kms^-1) retrieved from the literature. This bias will be corrected in the future data releases, when a common zero point for all the setups and instruments used for the survey will be established.
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