The extensive span of the Sagittarius (Sgr) stream makes it a promising tool for studying the gravitational potential of the Milky Way (MW). Characterizing its stellar kinematics can constrain halo properties and provide a benchmark for the paradigm of galaxy formation from cold dark matter. Accurate models of the disruption dynamics of the Sgr progenitor are necessary to employ this tool. Using a combination of analytic modeling and N-body simulations, we build a new model of the Sgr orbit and resulting stellar stream. In contrast to previous models, we simulate the full infall trajectory of the Sgr progenitor from the time it first crossed the MW virial radius 8 Gyr ago. An exploration of the parameter space of initial phase-space conditions yields tight constraints on the angular momentum of the Sgr progenitor. Our best-fit model is the first to accurately reproduce existing data on the 3D positions and radial velocities of the debris detected 100 kpc away in the MW halo. In addition to replicating the mapped stream, the simulation also predicts the existence of several arms of the Sgr stream extending to hundreds of kiloparsecs. The two most distant stars known in the MW halo coincide with the predicted structure. Additional stars in the newly predicted arms can be found with future data from the Large Synoptic Survey Telescope. Detecting a statistical sample of stars in the most distant Sgr arms would provide an opportunity to constrain the MW potential out to unprecedented Galactocentric radii.