We study the interaction of a supermassive black hole (SMBH) binary and a standard radiatively efficient thin accretion disc. We examine steady-state configurations of the disc and migrating SMBH system, self-consistently accounting for tidal and viscous torques and heating, radiative diffusion limited cooling, gas and radiation pressure, and the decay of the binary's orbit. We obtain a ‘phase diagram’ of the system as a function of binary parameters, showing regimes in which both the disc structure and migration have a different character. Although massive binaries can create a central gap in the disc at large radii, the tidal barrier of the secondary causes a significant pile-up of gas outside of its orbit, which can lead to the closing of the gap. We find that this spillover occurs at an orbital separation as large as gravitational radii, where M• = 107M7 M⊙ is the total binary mass. If the secondary is less massive than ∼106 M⊙, then the gap is closed before gravitational waves (GWs) start dominating the orbital decay. In this regime, the disc is still strongly perturbed, but the piled-up gas continuously overflows as in a porous dam, and crosses inside the secondary's orbit. The corresponding migration rate, which we label Type 1.5, is slower than the usual limiting cases known as Type I and II migration. Compared to an unperturbed disc, the steady-state disc in the overflowing regime is up to several hundred times brighter in the optical bands. Surveys such as PanSTARRS or LSST may discover the periodic variability of this population of binaries. Our results imply that the circumbinary discs around SMBHs can extend to small radii during the last stages of their merger, when they are detectable by LISA, and may produce coincident electromagnetic emission similar to active galactic nuclei.