The novel electronic properties of graphene1–4, 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 SiO2 layer supporting the graphene channels5,6. 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 mS mm21 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.