The internal rotation rates of the giant planets can be estimated by cloud motions, but such an approach is not very precise because absolute wind speeds are not known a priori and depend on latitude: periodicities in the radio emissions, thought to be tied to the internal planetary magnetic field, are used instead. Saturn, despite an apparently axisymmetric magnetic field, emits kilometre-wavelength (radio) photons from auroral sources. This emission is modulated at a period initially identified as 10 h 39 min 24 +/- 7 s, and this has been adopted as Saturn's rotation period. Subsequent observations, however, revealed that this period varies by +/-6 min on a timescale of several months to years. Here we report that the kilometric radiation period varies systematically by +/-1% with a characteristic timescale of 20-30 days. Here we show that these fluctuations are correlated with solar wind speed at Saturn, meaning that Saturn's radio clock is controlled, at least in part, by conditions external to the planet's magnetosphere. No correlation is found with the solar wind density, dynamic pressure or magnetic field; the solar wind speed therefore has a special function. We also show that the long-term fluctuations are simply an average of the short-term ones, and therefore the long-term variations are probably also driven by changes in the solar wind.