  Many astrophysical binaries, from planets to black holes, exert strong
torques on their circumbinary accretion disks, and are expected to
significantly modify the disk structure. Despite the several decade long
history of the subject, the joint evolution of the binary + disk system has not
been modeled with self-consistent assumptions for arbitrary mass ratios and
accretion rates. Here we solve the coupled binary-disk evolution equations
analytically in the strongly perturbed limit, treating the azimuthally-averaged
angular momentum exchange between the disk and the binary and the modifications
to the density, scale-height, and viscosity self-consistently, including
viscous and tidal heating, diffusion limited cooling, radiation pressure, and
the orbital decay of the binary. We find a solution with a central cavity and a
migration rate similar to those previously obtained for Type-II migration,
applicable for large masses and binary separations, and near-equal mass ratios.
However, we identify a distinct new regime, applicable at smaller separations
and masses, and mass ratio in the range 0.001< q < 0.1. For these systems, gas
piles up outside the binary's orbit, but rather than creating a cavity, it
continuously overflows as in a porous dam. The disk profile is intermediate
between a weakly perturbed disk (producing Type-I migration) and a disk with a
gap (with Type-II migration). However, the migration rate of the secondary is
typically slower than both Type-I and Type-II rates. We term this new regime
"Type-1.5" migration.
