We study circumbinary accretion discs in the framework of the restricted three-body problem (R3Bp) and via numerically solving the height-integrated equations of viscous hydrodynamics. Varying the mass ratio of the binary, we find a pronounced change in the behaviour of the disc near mass ratio q ≡ M_s/M_p ∼ 0.04. For mass ratios above q = 0.04, solutions for the hydrodynamic flow transition from steady, to strongly fluctuating; a narrow annular gap in the surface density around the secondary's orbit changes to a hollow central cavity; and a spatial symmetry is lost, resulting in a lopsided disc. This phase transition is coincident with the mass ratio above which stable orbits do not exist around the L4 and L5 equilibrium points of the R3Bp. Using the disco code, we find that for thin discs, for which a gap or cavity can remain open, the mass ratio of the transition is relatively insensitive to disc viscosity and pressure. The q = 0.04 transition has relevance for the evolution of massive black hole binary+disc systems at the centres of galactic nuclei, as well as for young stellar binaries and possibly planets around brown dwarfs.