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Supermassive black holes (SMBHs) are a ubiquitous component of the nuclei of galaxies. It is normally assumed that after the merger of two massive galaxies, a SMBH binary will form, shrink because of stellar or gas dynamical processes, and ultimately coalesce by emitting a burst of gravitational waves. However, so far it has not been possible to show how two SMBHs bind during a galaxy merger with gas because of the difficulty of modeling a wide range of spatial scales. Here we report hydrodynamical simulations that track the formation of a SMBH binary down to scales of a few light years after the collision between two spiral galaxies. A massive, turbulent, nuclear gaseous disk arises as a result of the galaxy merger. The black holes form an eccentric binary in the disk in less than 1 million years as a result of the gravitational drag from the gas rather than from the stars.