Halide double perovskites have gained significant attention, owing to their composition of low-toxicity elements, stability in air and long charge-carrier lifetimes. However, most double perovskites, including Cs2AgBiBr6, have wide bandgaps, which limits photoconversion efficiencies. The bandgap can be reduced through alloying with Sb3+, but Sb-rich alloys are difficult to synthesize due to the high formation energy of Cs2AgSbBr6, which itself has a wide bandgap. We develop a solution-based route to synthesize phase-pure Cs2Ag(SbxBi1-x)Br6 thin films, with the mixing parameter x continuously varying over the entire composition range. We reveal that the mixed alloys (x between 0.5 and 0.9) demonstrate smaller bandgaps than the pure Sb- and Bi-based compounds. The reduction in the bandgap of Cs2AgBiBr6 achieved through alloying (170 meV) is larger than if the mixed alloys had obeyed Vegard's law (70 meV). Through in-depth computations, we propose that bandgap lowering arises from the type II band alignment between Cs2AgBiBr6 and Cs2AgSbBr6. The energy mismatch between the Bi and Sb s and p atomic orbitals, coupled with their non-linear mixing, results in the alloys adopting a smaller bandgap than the pure compounds. Our work demonstrates an approach to achieve bandgap reduction and highlights that bandgap bowing may be found in other double perovskite alloys by pairing together materials forming a type II band alignment.