For the past 150 years, the prevailing view of the local interstellar medium has been based on a peculiarity known as the Gould Belt^1-4, an expanding ring of young stars, gas and dust, tilted about 20 degrees to the Galactic plane. However, the physical relationship between local gas clouds has remained unknown because the accuracy in distance measurements to such clouds is of the same order as, or larger than, their sizes^5-7. With the advent of large photometric surveys⁸ and the astrometric survey⁹, this situation has changed¹⁰. Here we reveal the three-dimensional structure of all local cloud complexes. We find a narrow and coherent 2.7-kiloparsec arrangement of dense gas in the solar neighbourhood that contains many of the clouds thought to be associated with the Gould Belt. This finding is inconsistent with the notion that these clouds are part of a ring, bringing the Gould Belt model into question. The structure comprises the majority of nearby star-forming regions, has an aspect ratio of about 1:20 and contains about three million solar masses of gas. Remarkably, this structure appears to be undulating, and its three-dimensional shape is well described by a damped sinusoidal wave on the plane of the Milky Way with an average period of about 2 kiloparsecs and a maximum amplitude of about 160 parsecs.