The interaction of the solar wind with Earth's magnetosphere gives rise to the bright polar aurorae and to geomagnetic storms, but the relation between the solar wind and the dynamics of the outer planets' magnetospheres is poorly understood. Jupiter's magnetospheric dynamics and aurorae are dominated by processes internal to the jovian system, whereas Saturn's magnetosphere has generally been considered to have both internal and solar-wind-driven processes. This hypothesis, however, is tentative because of limited simultaneous solar wind and magnetospheric measurements. Here we report solar wind measurements, immediately upstream of Saturn, over a one-month period. When combined with simultaneous ultraviolet imaging we find that, unlike Jupiter, Saturn's aurorae respond strongly to solar wind conditions. But in contrast to Earth, the main controlling factor appears to be solar wind dynamic pressure and electric field, with the orientation of the interplanetary magnetic field playing a much more limited role. Saturn's magnetosphere is, therefore, strongly driven by the solar wind, but the solar wind conditions that drive it differ from those that drive the Earth's magnetosphere.