Results from exoplanet surveys indicate that small planets (super-Earth size and below) are abundant in our Galaxy. However, little is known about their interiors and atmospheres. There is therefore a need to find small planets transiting bright stars, which would enable a detailed characterisation of this population of objects. We present the results of a search for the transit of the Earth-mass exoplanet Alpha Centauri Bb with the Hubble Space Telescope (HST). We observed Alpha Centauri B twice in 2013 and 2014 for a total of 40 hours. We achieve a precision of 115 ppm per 6-s exposure time in...