Aims: The nature of the thick disc and its relation to the thin disc is presently an important subject of debate. In fact, the structural and chemo-dynamical transition between disc populations can be used as a test of the proposed models of Galactic disc formation and evolution. Methods: We used the atmospheric parameters, [α/Fe] abundances, and radial velocities, which were determined from the Gaia-ESO Survey GIRAFFE spectra of FGK-type stars (first nine months of observations) to provide a chemo-kinematical characterisation of the disc stellar populations. We focussed on a subsample of 1016 stars with high-quality parameters, covering the volume | Z | < 4.5 kpc and R in the range 2-13 kpc. Results: We have identified a thin to thick disc separation in the [α/Fe] vs. [M/H] plane, thanks to the presence of a low-density region in the number density distribution. The thick disc stars seem to lie in progressively thinner layers above the Galactic plane, as metallicity increases and [α/Fe] decreases. In contrast, the thin disc population presents a constant value of the mean distance to the Galactic plane at all metallicities. In addition, our data confirm the already known correlations between Vφ and [M/H] for the two discs. For the thick disc sequence, a study of the possible contamination by thin disc stars suggests a gradient up to 64 ± 9 km s-1 dex-1. The distributions of azimuthal velocity, vertical velocity, and orbital parameters are also analysed for the chemically separated samples. Concerning the gradients with galactocentric radius, we find, for the thin disc, a flat behaviour of the azimuthal velocity, a metallicity gradient equal to -0.058 ± 0.008 dex kpc-1 and a very small positive [α/Fe] gradient. For the thick disc, flat gradients in [M/H] and [α/Fe] are derived. Conclusions: Our chemo-kinematical analysis suggests a picture where the thick disc seems to have experienced a settling process, during which its rotation increased progressively and, possibly, the azimuthal velocity dispersion decreased. At [M/H] ≈ -0.25 dex and [α/Fe]≈ 0.1 dex, the mean characteristics of the thick disc in vertical distance to the Galactic plane, rotation, rotational dispersion, and stellar orbits' eccentricity agree with that of the thin disc stars of the same metallicity, suggesting a possible connection between these two populations at a certain epoch of the disc evolution. Finally, the results presented here, based only on the first months of the Gaia ESO Survey observations, confirm how crucial large high-resolution spectroscopic surveys outside the solar neighbourhood are today for our understanding of the Milky Way history. Based on observations collected with the FLAMES spectrograph at the VLT/UT2 telescope (Paranal Observatory, ESO, Chile), for the Gaia-ESO Large Public Survey, programme 188.B-3002.Full Table 1 is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/567/A5