  Non-local thermodynamic equilibrium (NLTE) line formation for neutral and
singly-ionized iron is considered through a range of stellar parameters
characteristic of cool stars. A comprehensive model atom for Fe I and Fe II is
presented. Our NLTE calculations support the earlier conclusions that the
statistical equilibrium (SE) of Fe I shows an underpopulation of Fe I terms.
However, the inclusion of the predicted high-excitation levels of Fe I in our
model atom leads to a substantial decrease in the departures from LTE. As a
test and first application of the Fe I/II model atom, iron abundances are
determined for the Sun and four selected stars with well determined stellar
parameters and high-quality observed spectra. Within the error bars, lines of
Fe I and Fe II give consistent abundances for the Sun and two metal-poor stars
when inelastic collisions with hydrogen atoms are taken into account in the SE
calculations. For the close-to-solar metallicity stars Procyon and $\beta$ Vir,
the difference (Fe II - Fe I) is about 0.1 dex independent of the line
formation model, either NLTE or LTE. We evaluate the influence of departures
from LTE on Fe abundance and surface gravity determination for cool stars.
