The origin of Saturn's rings has not been adequately explained. The current rings are more than 90 to 95 per cent water ice, which implies that initially they were almost pure ice because they are continually polluted by rocky meteoroids. In contrast, a half-rock, half-ice mixture (similar to the composition of many of the satellites in the outer Solar System) would generally be expected. Previous ring origin theories invoke the collisional disruption of a small moon, or the tidal disruption of a comet during a close passage by Saturn. These models are improbable and/or struggle to account for basic properties of the rings, including their icy composition. Saturn has only one large satellite, Titan, whereas Jupiter has four large satellites; additional large satellites probably existed originally but were lost as they spiralled into Saturn. Here I report numerical simulations of the tidal removal of mass from a differentiated, Titan-sized satellite as it migrates inward towards Saturn. Planetary tidal forces preferentially strip material from the satellite's outer icy layers, while its rocky core remains intact and is lost to collision with the planet. The result is a pure ice ring much more massive than Saturn's current rings. As the ring evolves, its mass decreases and icy moons are spawned from its outer edge with estimated masses consistent with Saturn's ice-rich moons interior to and including Tethys.