  Supermassive black holes (SMBHs) are ubiquitous in galaxies with a sizable
mass. It is expected that a pair of SMBHs originally in the nuclei of two
merging galaxies would form a binary and eventually coalesce via a burst of
gravitational waves. So far theoretical models and simulations have been unable
to predict directly the SMBH merger timescale from ab-initio galaxy formation
theory, focusing only on limited phases of the orbital decay of SMBHs under
idealized conditions of the galaxy hosts. The predicted SMBH merger timescales
are long, of order Gyrs, which could be problematic for future gravitational
wave searches. Here we present the first multi-scale $\Lambda$CDM cosmological
simulation that follows the orbital decay of a pair of SMBHs in a merger of two
typical massive galaxies at $z\sim3$, all the way to the final coalescence
driven by gravitational wave emission. The two SMBHs, with masses $\sim10^{8}$
M$_{\odot}$, settle quickly in the nucleus of the merger remnant. The remnant
is triaxial and extremely dense due to the dissipative nature of the merger and
the intrinsic compactness of galaxies at high redshift. Such properties
naturally allow a very efficient hardening of the SMBH binary. The SMBH merger
occurs in only $\sim10$ Myr after the galactic cores have merged, which is two
orders of magnitude smaller than the Hubble time.
