The fluidity of water in confined geometries is relevant to processes ranging from tribology to protein folding, and its molecular mobility in pores and slits has been extensively studied using a variety of approaches. Studies in which liquid flow is measured directly suggest that the viscosity of aqueous electrolytes confined to films of thickness greater than about 2-3 nm remains close to that in the bulk; this behaviour is similar to that of non-associative organic liquids confined to films thicker than about 7-8 molecular layers. Here we observe that the effective viscosity of water remains within a factor of three of its bulk value, even when it is confined to films in the thickness range 3.5 +/- 1 to 0.0 +/- 0.4 nm. This contrasts markedly with the behaviour of organic solvents, whose viscosity diverges when confined to films thinner than about 5-8 molecular layers. We attribute this to the fundamentally different mechanisms of solidification in the two cases. For non-associative liquids, confinement promotes solidification by suppressing translational freedom of the molecules; however, in the case of water, confinement seems primarily to suppress the formation of the highly directional hydrogen-bonded networks associated with freezing.