  The extensive span of the Sagittarius (Sgr) stream makes it a promising tool
for studying the Milky Way gravitational potential. Characterizing its stellar
kinematics can constrain halo properties and provide a benchmark for the Cold
Dark Matter galaxy formation paradigm. 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 Milky Way 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
reproduce accurately 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 Milky Way 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 Milky Way
potential out to unprecedented Galactocentric radii.
