It is thought that the Earth's outer core consists mainly of liquid iron and that the convection of this metallic liquid gives rise to the Earth's magnetic field. A full understanding of this convection is hampered, however, by uncertainty regarding the viscosity of the outer core. Viscosity estimates from various sources span no less than 12 orders of magnitude(1,2), and it seems unlikely that this uncertainty will be substantially reduced by experimental measurements in the near future. Here we present dynamical first-principles simulations of liquid iron which indicate that the viscosity of iron at core temperatures and pressures is at the low end of the range of previous estimates - roughly 10 times that of typical liquid metals at ambient pressure. This estimate supports the promotion commonly made in magnetohydrodynamic models that the outer core is an inviscid fluid(3-5) undergoing small-scale circulation and turbulent convection(6), rather than large-scale global circulation.