A new state-of-the-art thermoelectric material, Tl2Ag12Te7+delta, which possesses an extremely low thermal conductivity of about 0.25 W m(-1) K-1 and a high figure-of-merit of up to 1.1 at 525 K, was obtained using a conventional solid-state reaction approach. Its subcell is a variant of the Zr2Fe12P7 type, but ultimately its structure was refined as a composite structure of a Tl2Ag12Te6 framework and a linear Te atom chain running along the c axis. The super-space group of the framework was determined to be P6(3)(00 gamma)s with a = b = 11.438(1) angstrom, c = 4.6256(5) angstrom, and that of the Te chain substructure has the same a and b axes, but c = 3.212(1) angstrom, space group P6(00 gamma)s. The modulation leads to the formation of Te-2 and Te-3 fragments in this chain and a refined formula of Tl2Ag11.5Te7.4. The structure consists of a complex network of three-dimensionally connected AgTe4 tetrahedra forming channels filled with the Tl atoms. The electronic structures of four different models comprising different Te chains, Tl2Ag12Te7, Tl2Ag12Te7.33, and 2x Tl2Ag12Te7.5, were computed using the WIEN2k package. Depending on the Te content within the chain, the models are either semiconducting or metallic. Physical property measurements revealed semiconducting properties, with an ultralow thermal conductivity, and excellent thermoelectric properties at elevated temperatures.