Natural composite materials are renowned for their mechanical strength and toughness: despite being highly mineralized, with the organic component constituting not more than a few per cent of the composite material, the fracture toughness exceeds that of single crystals of the pure mineral by two to three orders of magnitude. The judicious placement of the organic matrix, relative to the mineral phase, and the hierarchical structural architecture extending over several distinct length scales both play crucial roles in the mechanical response of natural composites to external loads. Here we use transmission electron microscopy studies and beam bending experiments to show that the resistance of the shell of the conch Strombus gigas to catastrophic fracture can be understood quantitatively by invoking two energy-dissipating mechanisms: multiple microcracking in the outer layers at low mechanical loads, and crack bridging in the shell's tougher middle layers at higher loads. Both mechanisms are intimately associated with the so-called crossed lamellar microarchitecture of the shell, which provides for 'channel' cracking in the outer layers and uncracked structural features that bridge crack surfaces, thereby significantly increasing the work of fracture, and hence the toughness, of the material. Despite a high mineral content of about 99% (by volume) of aragonite, the shell of Strombus gigas can thus be considered a 'ceramic plywood' and can guide the biomimetic design of tough, lightweight structures.