Noble gas anions of general formula FNgBN(-) (Ng = He-Xe) have been investigated by MP2, coupled-cluster, and multireference-Cl calculations with correlation-consistent basis sets. These species reside in deep wells on the singlet potential energy surface and are thermodynamically stable with respect to the loss of F, F-, BN. and BN-. They are unstable with respect to Ng + FBN-, but at least for Ng = Ar, Kr, and Xe, the involved energy barriers are high enough to suggest their conceivable existence as metastable species. The stability of FNgBN- arises from the strong F--stabilization of the elusive NgBN. The character of the boron-noble gas bond passes from purely ionic for FHeBN- and FNeBN- to covalent for FXeBN-.