We investigate the consequences of dimensional confinement on halide double perovskites by synthesizing two-dimensional analogues of the recently reported three-dimensional double perovskite Cs2AgBiBr6. The layered perovskites (BA)4AgBiBr8 (1) and (BA)2CsAgBiBr7 (2) (BA = CH3(CH2)3NH3+) feature metal-halide sheets of mono and bilayer thickness, respectively, where the ordered double-perovskite lattice is partitioned by organic cations. Electronic structure calculations indicate that the indirect bandgap of Cs2AgBiBr6 becomes direct when the infinitely thick inorganic lattice is reduced to monolayer thickness. Calculations on model systems allow us to separate the effects of dimensional reduction from those of the accompanying structural distortions in the inorganic sublattice. Detailed optical characterization shows that the photophysical properties of 1 and 2 are markedly different than those of their well-studied lead-halide analogs. Hybrid layered derivatives of double perovskites substantially expand on the substitutional flexibility of halide perovskites to encompass greater compositional and electronic diversity.