Spatiotemporal chaos (STC) near the onset of electroconvection in a nematic liquid crystal is reported. In samples with conductivities greater than 1 x 10(-8) per ohm per meter, STC was found to evolve by means of a supercritical Hopf bifurcation from the uniform conduction state. Because this example of STC resulted from nonlinear interactions between only four modes, it provides a realistic opportunity to understand the observed phenomena in terms of a weakly nonlinear theory in the form of four coupled complex Ginzburg-Landau equations derived from the full equations of motion of the system. For smaller conductivities, the pattern immediately above onset consisted of localized pulses of convection that coexisted with the conduction state. The pulses had a unique width in the direction perpendicular to the director (the axis parallel to the average orientation) and had much larger and varying lengths parallel to the director.