For four decades fairbankite was reported to have the formula Pb2+(Te4+O3), but repeated attempts to isolate fairbankite crystals for structural determination found only the visually similar cerussite and, more rarely, anglesite. The crystal-structure determination of fairbankite using single-crystal X-ray diffraction, supported by electron microprobe analysis and X-ray powder diffraction on the type specimen, has shown that fairbankite contains essential S, along with Pb, Te, and O. The chemical formula of fairbankite has been revised to (Te4+O3)11(SO4). This change has been accepted by the IMA–CNMNC, Proposal 19-I. The crystal structure of fairbankite [space group P1 (no. 1); revised cell: a = 7.0205(3) Å, b = 10.6828(6) Å, c = 14.4916(8) Å, a = 75.161(5)°, b = 81.571(4)°, g = 83.744(4)°, V = 1036.35(9) Å3, and Z = 1] is the first atomic arrangement known to contain a non-cyclic, finite building unit. Fairbankite has an average structure, formed from a 3D framework of Pb2+On polyhedra, Te4+On polyhedra, and SO4 tetrahedra in a 12:11:1 ratio. The stereoactive lone pairs of the Pb2+ and Te4+ cations are oriented into void space within the structure. Fairbankite contains two mixed sites statistically occupied by Te4+ and S6+ in approximately 4:1 and 1:4 ratios. These two sites possess Te4+ in trigonal-pyramidal environment and S6+ in tetrahedral environment (with an additional O site to create tetrahedral SO4 shape for the S-dominant site). Six of the 10 fully occupied Te4+ sites have Te4+ in trigonal-pyramidal environment, while four have Te4+ at the center of highly distorted Te4+O4 disphenoids. The disphenoids allow for the creation of two dimeric units in addition to the trimeric unit, which contains two disphenoids. All linkage between disphenoids and trigonal pyramids is via corner-linking. Secondary connectivity is via long Te–O and Pb–O bonds.