We present the Forgotten Quadrant Survey (FQS), an ESO large project that used the 12m antenna of the Arizona Radio Observatory to map the Galactic Plane in the range 220deg$<l<$240deg and -2.5deg$<b<$0deg, both in $^{12}$CO(1-0) and $^{13}$CO(1-0), at a spectral resolution of 0.65 km s$^{-1}$ and 0.26 km s$^{-1}$. Our dataset allows us to easily identify how the molecular dense gas is organised at different spatial scales: from the giant clouds with their denser filamentary networks, down to the clumps and cores that host the newborn stars and to obtain reliable estimates of their key physical parameters. We present the first release of the FQS data and discuss their quality. Spectra with 0.65 km s$^{-1}$ velocity channels have a noise ranging from 0.8 K to 1.3 K for $^{12}$CO(1-0) and from 0.3 K to 0.6 K for $^{13}$CO(1-0). In this paper, we used the $^{12}$CO(1-0) spectral cubes to produce a catalogue of 263 molecular clouds. This is the first selfconsistent, statistical catalogue of molecular clouds of the outer Galaxy, obtained with a subarcminute spatial resolution and therefore able to detect not only the classical giant molecular clouds, but also the small clouds and to resolve the cloud structure at the subparsec scale up to a distance of a few kpc. We found two classes of objects: structures with size above a few parsecs that are typical molecular clouds and may be self-gravitating, and subparsec structures that cannot be in gravitational equilibrium and are likely transient or confined by external pressure. We used the ratio between the Herschel H$_2$ column density and the integrated intensity of the CO lines to calculate the CO conversion factor and we found mean values of (3.3$pm$1.4)$times 10^{20}$ cm$^{-2}$(K km s$^{-1})^{-1}$ and (1.2$pm$0.4)$times 10^{21}$ cm$^{-2}$(K km s$^{-1})^{-1}$, for $^{12}$CO(1-0) and $^{13}$CO(1-0), respectively.
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