Three-dimensional (3D) hybrid organic-inorganic lead halide perovskites (HOIPs) feature remarkable optoelectronic properties for solar energy conversion but suffer from long-standing issues of environmental stability and lead toxicity. Associated two-dimensional (2D) analogues are garnering increasing interest due to superior chemical stability, structural diversity, and broader property tunability. Toward lead-free 2D HOIPs, double perovskites (DPs) with mixed-valent dual metals are attractive. Translation of mixed-metal DPs to iodides, with their prospectively lower bandgaps, represents an important target for semiconducting halide perovskites, but has so far proven inaccessible using traditional spacer cations due to either intrinsic instability or formation of competing non-perovskite phases. Here, we demonstrate the first example of a 2D Ag-Bi iodide DP with a direct bandgap of 2.00(2) eV, templated by a layer of bifunctionalized oligothiophene cations, i.e., (bis-aminoethyl)bithiophene, through a collective influence of aromatic interactions, hydrogen bonding, bidentate tethering, and structural rigidity. Hybrid density functional theory calculations for the new material reveal a direct bandgap, consistent with the experimental value, and relatively flat band edges derived principally from Ag-d/I-p (valence band) and Bi-p/I-p (conduction band) states. This work opens up new avenues for exploring specifically designed organic cations to stabilize otherwise inaccessible 2D HOIPs with potential applications for optoelectronics.