The recent discovery of lead-free halide double perovskites with band gaps in the visible represents an important step forward in the design of environmentally friendly perovskite solar cells. Within this new family of semiconductors, Cs2BiAgCl6 and Cs2BiAgBr6 are stable compounds crystallizing in the elpasolite structure. Following the recent computational discovery and experimental synthesis of these compounds, a detailed investigation of their electronic properties is warranted in order to establish their potential as optoelectronic materials. In this work, we perform many-body perturbation theory calculations and obtain high accuracy band gaps for both compounds. In addition, we report on the synthesis of Cs2BiAgBr6 single crystals, which are stable in ambient conditions. From our complementary theoretical and experimental analysis, we are able to assign the indirect character of the band gaps and obtain both experimental and theoretical band gaps of these novel semiconductors that are in close agreement.