Superconductivity was discovered in the layered compound RbBi11/3Te6, featuring Bi vacancies and a narrow band gap of 0.25(2) eV at room temperature. A sharp superconducting transition at similar to 3.2 K was observed in polycrystalline ingots. The superconducting volume fraction of oriented single crystals is almost 100%, confirming bulk superconductivity. Systematic Se and Sb substitutions in RbBi11/3-ySbySexTe6-x, revealed a dependence of the superconducting transition on composition that can increase the T-c up to similar to 10%. The RbBi11/3Te6 system is the first member of the new homologous series Rb[Bi2n+11/3Te3n+6] with infinite Bi2Te3-like layers. The large degree of chemical tunability of the electronic structure of the homology via doping and/or substitution gives rise to a new family of superconductors.