We compute the expected low-frequency gravitational wave signal from coalescing massive black-hole (MBH) binaries at the centres of galaxies. We follow the merging history of halos and associated holes via cosmological Monte Carlo realizations of the merger hierarchy from early times to the present in a ΛCDM cosmology. MBHs get incorporated through a series of mergers into larger and larger halos, sink to the centre owing to dynamical friction, accrete a fraction of the gas in the merger remnant to become more massive, and form a binary system. Stellar dynamical processes dominate the orbital evolution of the binary at large separations, while gravitational wave emission takes over at small radii, causing the final coalescence of the system. We discuss the observability of inspiralling MBH binaries by a low-frequency gravitational wave experiment such as the planned Laser Interferometer Space Antenna (LISA), discriminating between resolvable sources and unresolved confusion noise. Over a three-year observing period LISA should resolve this GWB into discrete sources, detecting ≈90 individual events above a S/N = 5 confidence level, while expected confusion noise is well below planned LISA capabilities.