We have indented the surface of ice at temperatures between Ϫ1°C and Ϫ17°C with sharp atomic force microscope tips. For a thick viscous interfacial melt layer, a Newtonian treatment of the flow of quasiliquid between the tip and the ice suggests that indentations at different indentation velocities should have the same force/velocity ratio for a given pit depth. This is observed for silicon tips with and without a hydrophobic coating at temperatures between Ϫ1°C and Ϫ10°C implying the presence of a liquid-like layer at the interface between tip and ice. At temperatures below about Ϫ10°C the dependence of force on velocity is weaker, suggesting that plastic flow of the ice dominates. A simple model for viscous flow that incorporates the approximate shape of our tip is used to obtain an estimate of the layer thickness, assuming the layer has the viscosity of supercooled water. The largest layer thicknesses inferred from this model are too thin to be described by continuum mechanics, but the model fits the data well. This suggests that the viscosity of the confined quasiliquid is much greater than that of bulk supercooled water. The hydrophobically coated tip has a significantly thinner layer than the uncoated tip, but the dependence of thickness on temperature is similar. The estimated viscous layer thickness increases with increasing temperature as expected for a quasiliquid premelt layer.