Most of the old cratered highlands of Mars are dissected by branching river valleys that appear to have been cut by running water, yet liquid water is unstable everywhere on the Martian surface. In the equatorial region, where most of the valleys are observed, even ice is unstable. It has been suggested, therefore, that Mars had an early denser atmosphere with sufficient greenhouse warming to allow the existence of liquid water. Here, it is suggested instead that during periods of very high obliquities, ice could accumulate at low latitudes as a result of sustained sublimation of ice from the poles and transport of the water vapor equatorwards. At low latitudes, the water vapor would saturate the atmosphere and condense onto the surface, where it would accumulate until lower obliquities prevailed. The mechanism is efficient only at the very high obliquities that occurred before formation of Tharsis very early in the planet's history, but limited equatorial ice accumulation could also have occurred at the highest obliquities during the rest of the planet's history. Partial melting of the ice could have provided runoff to form the channels or replenish the groundwater system.