The novel electronic properties of graphene(1-4), including a linear energy dispersion relation and purely two-dimensional structure, have led to intense research into possible applications of this material in nanoscale devices. Here we report the first observation of saturating transistor characteristics in a graphene field-effect transistor. The saturation velocity depends on the charge-carrier concentration and we attribute this to scattering by interfacial phonons in the SiO(2) layer supporting the graphene channels(5,6). Unusual features in the current-voltage characteristic are explained by a field-effect model and diffusive carrier transport in the presence of a singular point in the density of states. The electrostatic modulation of the channel through an efficiently coupled top gate yields transconductances as high as 150 mu S mu m(-1) despite low on-off current ratios. These results demonstrate the feasibility of two-dimensional graphene devices for analogue and radio-frequency circuit applications without the need for bandgap engineering.