Saturn's rings are rock-poor, containing 90 to 95% ice by mass. As a group, Saturn's moons interior to and including Tethys are also about 90% ice. Tethys itself contains < 6% rock by mass, in contrast to its similar-mass outer neighbor Dione, which contains > 40% rock. Here we simulate the evolution of a massive primordial ice-rich ring and the production of satellites as ring material spreads beyond the Roche limit. We describe the Roche-interior ring with an analytic model, and use an N-body code to describe material beyond the Roche limit. We track the accretion and interactions of spawned satellites, including tidal interaction with the planet, assuming a tidal dissipation factor for Saturn of Q ~ 104. We find that ring torques and capture of moons into mutual resonances produces a system of ice-rich inner moons that extends outward to approximately Tethys's orbit in 109 years, even with relatively slow orbital expansion due to tides. The resulting mass and semi-major axis distribution of spawned moons resembles that of Mimas, Enceladus and Tethys. We estimate the mass of rock delivered to the moons by external cometary impactors during a late-heavy bombardment. We find that the inner moons receive a mass in rock comparable to their current total rock content, while Dione and Rhea receive an order-of-magnitude less rock than their current rock content. This suggests that external contamination may have been the primary source of rock in the inner moons, and that Dione and Rhea formed from much more rock-rich source material. Reproducing the distribution of rock among the current inner moons is challenging, and appears to require large impactors and stochasticity and/or the presence of some rock in the initial ring.