For the past 150 years, the prevailing view of the local interstellar medium was based on a peculiarity known as Gould’s Belt1–4, an expanding ring of young stars, gas and dust, tilted about 20 degrees to the Galactic plane. Still, the physical relation between local gas clouds has remained practically unknown because the distance accuracy to clouds is of the same order as, or larger than, their sizes5–7. With the advent of large photometric surveys8 and astrometric survey9 this situation has changed10. Here we report 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, disputing the Gould Belt model. The new 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, the new structure appears to be undulating and its three-dimensional structure 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.