Solitary waves have, for the first time, been identified in 3D electric field data at the subsolar, equatorial magnetopause. These nonlinear, bipolar electric field pulses parallel to the magnetic field occur both as individual spikes and as trains of spikes. The solitary waves have amplitudes up to ~25 mV/m, and velocities from ~150 km/s to >2000 km/s, with scale sizes the order of a kilometer (comparable to the Debye length). Almost all the observed solitary waves are positive potential structures with potentials of ~0.1 to 5 Volts. They are often associated with very large amplitude waves in either or both the electric and magnetic fields. Although most of the observed signatures are consistent with an electron hole mode, the events with very low velocities and the few negative potential structures may be indicative of a second type of solitary wave in the magnetopause current layer. The solitary waves may be an important source of dissipation and diffusion at the magnetopause.