The nearby transiting planet HD 189733b was observed during three transits with the Advanced Camera for Surveys of the Hubble Space Telescope in spectroscopic mode. The resulting time-series of 675 spectra covers the 550–1050 nm range, with a resolution element of ∼8 nm, at extremely high accuracy (signal-to-noise ratio up to 10 000 in 50-nm intervals in each individual spectrum). Using these data, we disentangle the effects of limb darkening, measurement systematics and spots on the surface of the host star, to calculate the wavelength dependence of the effective transit radius to an accuracy of ∼50 km. This constitutes the ‘transmission spectrum’ of the planetary atmosphere. It indicates at each wavelength at what height the planetary atmosphere becomes opaque to the grazing stellar light during the transit. In this wavelength range, strong features due to sodium, potassium and water are predicted by atmosphere models for a planet like HD 189733b, but they can be hidden by broad absorption from clouds or hazes higher up in the atmosphere. We observed an almost featureless transmission spectrum between 550 and 1050 nm, with no indication of the expected sodium or potassium atomic absorption features. Comparison of our results with the transit radius observed in the near and mid-infrared (2–8 μm), and the slope of the spectrum, suggest the presence of a haze of submicrometre particles in the upper atmosphere of the planet.