Most short period transiting exoplanets have circular orbits, as expected from an estimation of the circularisation timescale using classical tidal theory. Interestingly, a small number of short period transiting exoplanets seem to have orbits with a small eccentricity. Such systems are valuable as they may indicate that some key physics is missing from formation and evolution models. We have analysed the results of a campaign of radial velocity measurements of known transiting planets with the SOPHIE and HARPS spectrographs using Bayesian methods and obtained new constraints on the orbital elements of 12 known transiting exoplanets. We also reanalysed the radial velocity data for another 42 transiting systems and show that some of the eccentric orbits reported in the Literature are compatible with a circular orbit. As a result, we show that the systems with circular and eccentric orbits are clearly separated on a plot of the planetary mass versus orbital period. We also show that planets following the trend where heavier hot Jupiters have shorter orbital periods (the “mass-period relation” of hot Jupiters), also tend to have circular orbits, with no confirmed exception to this rule so far.