We used the database of 1040 short-period (1<=P<200days) exoplanets radial-velocity orbits to study the planetary eccentricity-period (PEP) distribution. We first divided the sample into low- and high-mass exoplanet subsamples based on the distribution of the (minimum) planetary masses, which displays a clear two-Gaussian distribution, separated at 0.165M_J_. We then selected 216 orbits, low- and high-mass alike, with eccentricities significantly distinct from circular orbits. The 131 giant-planet eccentric orbits display a clear upper envelope, which we model quantitatively, rises monotonically from zero eccentricity and reaches an eccentricity of 0.8 at P~100 days. Conversely, the 85 low-mass planetary orbits display a flat eccentricity distribution between 0.1 and 0.5, with almost no dependence on the orbital period. We show that the striking difference between the two PEP distributions is not a result of the detection technique used. The upper envelope of the high-mass planets, also seen in short-period binary stars, is a clear signature of tidal circularization, which probably took place inside the planets, while the small-planet PEP distribution suggests that the circularization was not effective, probably due to dynamical interactions with neighboring planets.