Context : Aims : 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 220° < l < 240° and -2 \aas@@fstack { \circ } 5 < b < 0° , 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 } . Methods : We used the ( 1–0 ) transition of carbon monoxide to trace the molecular component of the interstellar medium . Our data set 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 new-born stars and to obtain reliable estimates of their key physical parameters such as size and mass . Results : We present the first release of the data of the FQS survey and discuss their quality . Spectra with 0.65 km s ^ { -1 } velocity channels have 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 first paper , we used the ^ { 12 } CO ( 1–0 ) spectral cubes to produce a catalogue of 263 molecular clouds . The clouds are grouped in three main structures corresponding to the Local , Perseus , and Outer arms up to a distance of \sim 8.6 kpc from the Sun . This is the first self-consistent 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 sub-parsec scale up to a distance of a few kiloparsec . We found two classes of objects : structures with sizes above a few parsecs that are typical molecular clouds and may be self-gravitating , and subparsec structures that can not 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 . Conclusions : The FQS contributes to the general effort in producing a new generation of high-quality spectroscopic data for the Galactic plane in the less-studied third Galactic quadrant toward the outer Galaxy . The FQS has produced a data set of great legacy value , largely improving the data quality both in terms of sensitivity and spatial resolution over previous data sets .