/ AbstractMeV-GeV dark matter (DM) is theoretically well motivated but remarkably unexplored. This proposal presents the MeV-GeV DM discovery potential for a $\sim$1 m$^3$ segmented CsI(Tl) scintillator detector placed downstream of the Hall A beam-dump at Jefferson Lab, receiving up to 10$^{22}$ electrons-on-target (EOT) in 285 days. This experiment (Beam-Dump eXperiment or BDX) would be sensitive to elastic DM-electron and to inelastic DM scattering at the level of 10 counts per year, reaching the limit of the neutrino irreducible background. The distinct signature of a DM interaction will be an electromagnetic shower of few hundreds of MeV, together with a reduced activity in the surrounding active veto counters. A detailed description of the DM particle $χ$ production in the dump and subsequent interaction in the detector has been performed by means of Monte Carlo simulations. Different approaches have been used to evaluate the expected backgrounds: the cosmogenic background has been extrapolated from the results obtained with a prototype detector running at INFN-LNS (Italy), while the beam-related background has been evaluated by GEANT4 Monte Carlo simulations. The proposed experiment will be sensitive to large regions of DM parameter space, exceeding the discovery potential of existing and planned experiments in the MeV-GeV DM mass range by up to two orders of magnitude.