Many astrophysical binaries, from planets to black holes, exert strong torques on their circumbinary accretion discs, and are expected to significantly modify the disc structure. Despite the several decade long history of the subject, the joint evolution of the binary + disc system has not been modelled with self-consistent assumptions for arbitrary mass ratios and accretion rates. Here, we solve the coupled binary–disc evolution equations analytically in the strongly perturbed limit, treating the azimuthally averaged angular momentum exchange between the disc and the binary and the modifications to the density, scaleheight, 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 10−3 ≲ 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 disc profile is intermediate between a weakly perturbed disc (producing Type I migration) and a disc 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.