  The influence of confinement, due to flat parallel structureless walls, on
phase separation in colloid-polymer mixtures, is investigated by means of
grand-canonical Monte Carlo simulations. Ultra-thin films, with thicknesses
between $D=3-10$ colloid diameters, are studied. The Asakura-Oosawa model [J.
Chem. Phys. 22, 1255 (1954)] is used to describe the particle interactions. To
simulate efficiently, a ``cluster move'' [J. Chem. Phys. 121, 3253 (2004)] is
used in conjunction with successive umbrella sampling [J. Chem. Phys. 120,
10925 (2004)]. These techniques, when combined with finite size scaling, enable
an accurate determination of the unmixing binodal. Our results show that the
critical behavior of the confined mixture is described by ``effective''
critical exponents, which gradually develop from values near those of the
two-dimensional Ising model, to those of the three-dimensional Ising model, as
$D$ increases. The scaling predictions of Fisher and Nakanishi [J. Chem. Phys.
75, 5875 (1981)] for the shift of the critical point are compatible with our
simulation results. Surprisingly, however, the colloid packing fraction at
criticality approaches its bulk ($D \to \infty$) value non-monotonically, as
$D$ is increased. Far from the critical point, our results are compatible with
the simple Kelvin equation, implying a shift of order 1/D in the coexistence
colloid chemical potential. We also present density profiles and pair
distribution functions for a number of state points on the binodal, and the
influence of the colloid-wall interaction is studied.
