We have conducted a spectroscopic survey of the inner regions of the Sagittarius (Sgr) dwarf galaxy using the AAOmega spectrograph on the Anglo-Australian Telescope. We determine radial velocities for over 1800 Sgr star members in six fields that cover an area 18.84 deg2, with a typical accuracy of 2 km s-1. Motivated by recent numerical models of the Sgr tidal stream that predict a substantial amount of rotation in the dwarf remnant core, we compare the kinematic data against N-body models that simulate the stream progenitor as (1) a pressure-supported, mass-follows-light system and (2) a late-type, rotating disk galaxy embedded in an extended dark matter halo. We find that the models with little or no intrinsic rotation clearly yield a better match to the mean line-of-sight velocity in all surveyed fields, but fail to reproduce the shape of the line-of-sight velocity distribution. This result rules out models wherein the prominent bifurcation observed in the leading tail of the Sgr stream was caused by a transfer from intrinsic angular momentum from the progenitor satellite into the tidal stream. It also implies that the trajectory of the young tidal tails has not been affected by internal rotation in the progenitor system. Our finding indicates that new, more elaborate dynamical models, in which the dark and luminous components are treated independently, are necessary for simultaneously reproducing both the internal kinematics of the Sgr dwarf and the available data for the associated tidal stream.