  We present a new technique which allows the fully {\em ab initio} calculation
of the chemical potential of a substitutional impurity in a high-temperature
crystal, including harmonic and anharmonic lattice vibrations. The technique
uses the combination of thermodynamic integration and reference models
developed recently for the {\em ab initio} calculation of the free energy of
liquids and anharmonic solids. We apply the technique to the case of the
substitutional oxygen impurity in h.c.p. iron under Earth's core conditions,
which earlier static {\em ab initio} calculations indicated to be
thermodynamically very unstable. Our results show that entropic effects arising
from the large vibrational amplitude of the oxygen impurity give a major
reduction of the oxygen chemical potential, so that oxygen dissolved in h.c.p.
iron may be stabilised at concentrations up a few mol % under core conditions.
