A hollow aluminum structure that will be brazed includes at least one brazing sheet having a filler metal layer clad onto a core layer. The core layer is composed of aluminum or an aluminum alloy containing less than 0.2 mass % Mg. The filler metal layer is composed of an aluminum alloy that contains Si: 4.0-13.0 mass % and Bi: 0.01-0.3 mass %, and further contains Li: 0.004-0.08 mass % and/or Be: 0.006-0.12 mass %, the filler metal layer containing less than 0.1 mass % Mg. The hollow aluminum structure is assembled such that the filler metal layer is present at locations that will form both an interior-facing brazed joint and an exterior-facing brazed joint. Then, flux is applied onto the filler metal layer at the location that will form the exterior brazed joint, and the hollow aluminum structure heated in an inert gas atmosphere to form the interior brazed joint and the exterior brazed joint.