(Abridged) We have used the atmospheric parameters, [alpha/Fe] abundances and radial velocities, determined from the Gaia-ESO Survey GIRAFFE spectra of FGK-type stars (iDR1), to provide a chemo-kinematical characterisation of the disc stellar populations. We focuss on a subsample of 1016 stars with high quality parameters, covering the volume |Z|<4.5kpc and R in the range 2-13kpc. We have identified a thin to thick disc separation in the [alpha/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 [alpha/Fe] decreases. The thin disc population presents a constant value of the mean distance to the plane at all metallicities. Our data confirm the already known correlations between V_phi 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 64km/s/dex. The distributions of V_phi, V_Z, and orbital parameters are analysed for the chemically separated samples. Concerning the gradients with galactocentric radius, we find for the thin disc a flat behaviour of V_phi, a [M/H] gradient of -0.058dex/kpc and a small positive [alpha/Fe] gradient. For the thick disc, flat gradients in [M/H] and [alpha/Fe] are derived. Our chemo-kinematical analysis suggests a picture in which the thick disc seems to have experienced a settling process, during which its rotation increased progressively, and, possibly, the V_phi dispersion decreased. At [M/H]-0.25dex and [alpha/Fe]0.1dex, the mean characteristics of the thick disc in distance to the Galactic plane, V_phi, V_phi dispersion and eccentricity agree with those of the thin disc stars, suggesting a possible connection between these populations at a certain epoch of the disc evolution.
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