The free energy and local height probabilities of the dilute A models with
broken Z2 symmetry are calculated analytically using inversion and corner
transfer matrix methods. These models possess four critical branches. The first
two branches provide new realizations of the unitary minimal series and the
other two branches give a direct product of this series with an Ising model. We
identify the integrable perturbations which move the dilute A models away from
the critical limit. Generalized order parameters are defined and their critical
exponents extracted. The associated conformal weights are found to occur on
the diagonal of the relevant Kac table. In an appropriate regime the dilute A_3
model lies in the universality class of the lsing model in a magnetic field. In this
case we obtain the magnetic exponent 6 = 15 directly, without the use of scaling
relations.