The preparation of a new random intergrowth structure in the TlBaCaCuO system is reported. Metastable Tl1+xBa2Ca2Cu3O9+x thin films with composition intermediate between TlBa2Ca2Cu3O9 and Tl2Ba2Ca2Cu3O10 were fabricated through a two-step process involving deposition of polycrystalline BaCaCu oxide precursor films of composition Ba:Ca:Cu=1.76:2.04:3.20 onto pseudocubic (100) LaAlO3 at 500-degrees-C. followed by thallination in air at 880-degrees-C for 20 min in the presence of a controlled Ti2O partial pressure generated from a powder containing Tl2O3 and a mixture of T]BaCaCuO phases with nominal cation composition Tl: Ba: Ca: Cu is similar to 1.8: 2.0:2.0: 3.0. X-ray diffraction, electron diffraction, and high resolution electron microscopy analyses showed these films to consist of a nearly random mixture of roughly equal amounts of TlBa2Ca2Cu3O9 and Tl2Ba2Ca2Cu3O10 layers, rather than distinct TlBa2Ca2Cu3O9 and Tl2Ba2Ca2Cu3O10 phases, yielding a material of ideal stoichiometry TlBa2Ca2Cu3O9+, (x congruent-to 0.5). While this partially disordered layered material exhibits periodicity within the individual layers, it is characterized by an absence of periodicity perpendicular to these layers. The films' superconducting transition temperature T(c) for zero resistivity was 107.4 +/- 0.5 K, the inductively measured T(c) was 108 +/- 2 K, and the onset of flux exclusion as measured by AC magnetic susceptibility began at 111.5 +/- 0.5 K. The critical current density J(c) was measured to be 2.4 x 10(4) A/cm2 at 4.2 K and 3.4 X 10(3) A/cm2 at 80 K. The electrical measurements are indicative of a material containing high concentrations of weak links.