We measure the star formation quenching efficiency and time-scale in cluster environments. Our method uses N-body simulations to estimate the probability distribution of possible orbits for a sample of observed Sloan Digital Sky Survey galaxies in and around clusters based on their position and velocity offsets from their host cluster. We study the relationship between their star formation rates and their likely orbital histories via a simple model in which star formation is quenched once a delay time after infall has elapsed. Our orbit library method is designed to isolate the environmental effect on the star formation rate due to a galaxy's present-day host cluster from 'pre-processing' in previous group hosts. We find that quenching of satellite galaxies of all stellar masses in our sample ... by massive ... clusters is essentially 100 per cent efficient. Our fits show that all galaxies quench on their first infall, approximately at or within a Gyr of their first pericentric passage. There is little variation in the onset of quenching from galaxy-to-galaxy: the spread in this time is at most ~2 Gyr at fixed M*. Higher mass satellites quench earlier, with very little dependence on host cluster mass in the range probed by our sample. (ProQuest: ... denotes formulae/symbols omitted.)