  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 a
highly-saturated regime, which is found to be consistent across HST orbits. We
rule out the transiting nature of Alpha Centauri Bb with the orbital parameters
published in the literature at 96.6% confidence. We find in our data a single
transit-like event that could be associated to another Earth-size planet in the
system, on a longer period orbit. Our program demonstrates the ability of HST
to obtain consistent, high-precision photometry of saturated stars over 26
hours of continuous observations.
