Topological insulators are a new class of insulators in which a bulk gap for electronic excitations is generated because of the strong spin-orbit coupling inherent to these systems. These materials are distinguished from ordinary insulators by the presence of gapless metallic surface states, resembling chiral edge modes in quantum Hall systems, but with unconventional spin textures. A key predicted feature of such spin-textured boundary states is their insensitivity to spin-independent scattering, which is thought to protect them from backscattering and localization. Recently, experimental and theoretical efforts have provided strong evidence for the existence of both two- and three-dimensional classes of such topological insulator materials in semiconductor quantum well structures and several bismuth-based compounds, but so far experiments have not probed the sensitivity of these chiral states to scattering. Here we use scanning tunnelling spectroscopy and angle-resolved photoemission spectroscopy to visualize the gapless surface states in the three-dimensional topological insulator Bi(1-x)Sb(x), and examine in detail the influence of scattering from disorder caused by random alloying in this compound. We show that, despite strong atomic scale disorder, backscattering between states of opposite momentum and opposite spin is absent. Our observations demonstrate that the chiral nature of these states protects the spin of the carriers. These chiral states are therefore potentially useful for spin-based electronics, in which long spin coherence is critical, and also for quantum computing applications, where topological protection can enable fault-tolerant information processing.