The experimental realization of quantum spin liquids is a long-sought goal in physics, as they represent new states of matter. Quantum spin liquids cannot be described by the broken symmetries associated with conventional ground states. In fact, the interacting magnetic moments in these systems do not order, but are highly entangled with one another over long ranges. Spin liquids have a prominent role in theories describing high-transition-temperature superconductors, and the topological properties of these states may have applications in quantum information. A key feature of spin liquids is that they support exotic spin excitations carrying fractional quantum numbers. However, detailed measurements of these 'fractionalized excitations' have been lacking. Here we report neutron scattering measurements on single-crystal samples of the spin-1/2 kagome-lattice antiferromagnet ZnCu(3)(OD)(6)Cl(2) (also called herbertsmithite), which provide striking evidence for this characteristic feature of spin liquids. At low temperatures, we find that the spin excitations form a continuum, in contrast to the conventional spin waves expected in ordered antiferromagnets. The observation of such a continuum is noteworthy because, so far, this signature of fractional spin excitations has been observed only in one-dimensional systems. The results also serve as a hallmark of the quantum spin-liquid state in herbertsmithite.