We describe an experiment to search for a new vector boson A' with weak coupling alpha' > 6 x 10^{-8} alpha to electrons (alpha=e^2/4pi) in the mass range 65 MeV < m_A' < 550 MeV. New vector bosons with such small couplings arise naturally from a small kinetic mixing of the "dark photon" A' with the photon -- one of the very few ways in which new forces can couple to the Standard Model -- and have received considerable attention as an explanation of various dark matter related anomalies. A' bosons are produced by radiation off an electron beam, and could appear as narrow resonances with small production cross-section in the trident e+e- spectrum. We summarize the experimental approach described in a proposal submitted to Jefferson Laboratory's PAC35, PR-10-009. This experiment, the A' Experiment (APEX), uses the electron beam of the Continuous Electron Beam Accelerator Facility at Jefferson Laboratory (CEBAF) at energies of ~1-4 GeV incident on 0.5-10% radiation length Tungsten wire mesh targets, and measures the resulting e+e- pairs to search for the A' using the High Resolution Spectrometer and the septum magnet in Hall A. With a ~1 month run, APEX will achieve very good sensitivity because the statistics of e+e- pairs will be ~10,000 times larger in the explored mass range than any previous search for the A' boson. These statistics and the excellent mass resolution of the spectrometers allow sensitivity to alpha'/alpha one to three orders of magnitude below current limits, in a region of parameter space of great theoretical and phenomenological interest. Similar experiments could also be performed at other facilities, such as the Mainz Microtron. Show Partial Abstract