  The spatial variation of the colour of a galaxy may introduce a bias in the
measurement of its shape if the PSF profile depends on wavelength. We study how
this bias depends on the properties of the PSF and the galaxies themselves. The
bias depends on the scales used to estimate the shape, which may be used to
optimise methods to reduce the bias. Here we develop a general approach to
quantify the bias. Although applicable to any weak lensing survey, we focus on
the implications for the ESA Euclid mission.
  Based on our study of synthetic galaxies we find that the bias is a few times
10^-3 for a typical galaxy observed by Euclid. Consequently, it cannot be
neglected and needs to be accounted for. We demonstrate how one can do so using
spatially resolved observations of galaxies in two filters. We show that HST
observations in the F606W and F814W filters allow us to model and reduce the
bias by an order of magnitude, sufficient to meet Euclid's scientific
requirements. The precision of the correction is ultimately determined by the
number of galaxies for which spatially-resolved observations in at least two
filters are available. We use results from the Millennium Simulation to
demonstrate that archival HST data will be sufficient for the tomographic
cosmic shear analysis with the Euclid dataset.
