The relations between the specific angular momenta ($j$) and masses ($M$) of galaxies are often used as a benchmark in analytic models and hydrodynamical simulations as they are considered to be amongst the most fundamental scaling relations. Using accurate measurements of the stellar ($j_ast$), gas ($j_{rm gas}$), and baryonic ($j_{rm bar}$) specific angular momenta for a large sample of disc galaxies, we report the discovery of tight correlations between $j$, $M$, and the cold gas fraction of the interstellar medium ($f_{rm gas}$). At fixed $f_{rm gas}$, galaxies follow parallel power laws in 2D $(j,M)$ spaces, with gas-rich galaxies having a larger $j_ast$ and $j_{rm bar}$ (but a lower $j_{rm gas}$) than gas-poor ones. The slopes of the relations have a value around 0.7. These new relations are amongst the tightest known scaling laws for galaxies. In particular, the baryonic relation ($j_{rm bar}-M_{rm bar}-f_{rm gas}$), arguably the most fundamental of the three, is followed not only by typical discs but also by galaxies with extreme properties, such as size and gas content, and by galaxies previously claimed to be outliers of the standard 2D $j-M$ relations. The stellar relation ($j_{ast}-M_{ast}-f_{rm gas}$) may be connected to the known $j_ast-M_ast-$bulge fraction relation; however, we argue that the $j_{rm bar}-M_{rm bar}-f_{rm gas}$ relation can originate from the radial variation in the star formation efficiency in galaxies, although it is not explained by current disc instability models.
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