We report on high spectral and spatial resolution kinematic observations of H{alpha} emission from the Cartwheel system obtained with a scanning Fabry-Perot interferometer. Velocity fields for two galaxies were measured, for the main galaxy of the system, the Cartwheel galaxy, and for a spiral companion ~25h^-1^kpc to the north. The velocity field and the rotation curve of the spiral companion shows classical differential rotation and indicates the mass of the galaxy to be in the range 2.9-4.8x10^9^M{sun}. The Cartwheel galaxy contains strong H{alpha} in emission asymmetrically distributed along its outer ring, with some faint emission detected inside it. The velocity field and the rotation curve of the Cartwheel galaxy is clearly dominated by differential rotation although second-order effects cause distortions which can be understood in a scenario where expansion and density-wave motions are also considered. Our best model-fit to the observed velocity field of the Cartwheel galaxy includes three components for the kinematic of its outer ring: a circular velocity of 217km/s, an expansion velocity of 13-30+/-10km/s and a sinusoidal perturbation with an amplitude of 20+/-5km/s. This expansion velocity, lower than found in previous studies, implies that the ring could be older than previously thought. The rotation curve of the Cartwheel galaxy is consistent with a progenitor which had the morphology of a late-type spiral galaxy. Our measurements support the collisional models of ring galaxy formation and should impose strong constraints to simulations.