We are undertaking a large-scale radial velocity survey of the Galactic bulge that uses M giant stars selected from the Two Micron All Sky Survey catalog as targets for the Cerro Tololo Inter-American Observatory 4 m Hydra multiobject spectrograph. The aim of this survey is to test dynamical models of the bulge and to quantify the importance, if any, of cold stellar streams in the bulge and its vicinity. Here we report on the kinematics of a strip of fields at -10° < l < +10° and b = -4°. We construct a longitude-velocity plot for the bulge stars and the model data and find that, contrary to previous studies, the bulge does not rotate as a solid body. From -5° < l < +5° the rotation curve has a slope of roughly 100 km s-1 kpc-1 and flattens considerably at greater l, reaching a maximum rotation of 45 km s-1. We compare our rotation curve and velocity dispersion profile to both the self-consistent model of Zhao and to N-body models; neither fits both our observed rotation curve and velocity dispersion profile. The high precision of our radial velocities (~3 km s-1) yields an unexpected result: hints of cold kinematic features are seen in a number of the line-of-sight velocity distributions.