Context. In the past 15 years, several studies suggested that water in the stratosphere of Jupiter originated from the Shoemaker-Levy 9 (SL9) comet impacts in July 1994, but a direct proof was missing. Only a very sensitive instrument observing with high spectral/spatial resolution can help to solve this problem. This is the case of the Herschel Space Observatory, which is the first telescope capable of mapping water in Jupiter's stratosphere. Aims: We observed the spatial distribution of the water emission in Jupiter's stratosphere with the Heterodyne Instrument for the Far Infrared (HIFI) and the Photodetector Array Camera and Spectrometer (PACS) onboard Herschel to constrain its origin. In parallel, we monitored Jupiter's stratospheric temperature with the NASA Infrared Telescope Facility (IRTF) to separate temperature from water variability. Methods: We obtained a 25-point map of the 1669.9 GHz water line with HIFI in July 2010 and several maps with PACS in October 2009 and December 2010. The 2010 PACS map is a 400-point raster of the water 66.4 μm emission. Additionally, we mapped the methane ν4 band emission to constrain the stratospheric temperature in Jupiter in the same periods with the IRTF. Results: Water is found to be restricted to pressures lower than 2 mbar. Its column density decreases by a factor of 2-3 between southern and northern latitudes, consistently between the HIFI and the PACS 66.4 μm maps. We infer that an emission maximum seen around 15 °S is caused by a warm stratospheric belt detected in the IRTF data. Conclusions: Latitudinal temperature variability cannot explain the global north-south asymmetry in the water maps. From the latitudinal and vertical distributions of water in Jupiter's stratosphere, we rule out interplanetary dust particles as its main source. Furthermore, we demonstrate that Jupiter's stratospheric water was delivered by the SL9 comet and that more than 95% of the observed water comes from the comet according to our models. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.Figures 1 and 3 are available in electronic form at http://www.aanda.org< Réduire