This thesis describes the physics, design, and construction of an experiment to measure the electric dipole moment (EDM) of the electron. In the experiment, laser-cooled Cs atoms will be held in an optical dipole force trap in the presence of applied electric and magnetic fields. The signature of an electron EDM is a first-order electric field shift of the Zeeman resonance frequency of the Cs ground state. We present an analysis of the systematic and statistical errors of this experiment, which shows that the experiment should have a sensitivity of the order of 10⁻²⁹ e-cm. We pay particular attention to potential light-shift induced errors and to magnetic field noise. We also present the design and experimental results for a cold Cs atom source, high voltage field plates, optical trapping field in a resonant build-up cavity, noval titanim ultrahigh vacuum system, and magnetic sheilding system. These results show that a measurement of the electron edm at the level of 10⁻²⁹ e-cm. should be feasible. / text