We present a search at the Jefferson Laboratory for new forces mediated by sub-GeV vector bosons with weak coupling ${\ensuremath{\alpha}}^{\ensuremath{'}}$ to electrons. Such a particle ${A}^{\ensuremath{'}}$ can be produced in electron-nucleus fixed-target scattering and then decay to an ${e}^{+}{e}^{\ensuremath{-}}$ pair, producing a narrow resonance in the QED trident spectrum. Using APEX test run data, we searched in the mass range 175--250 MeV, found no evidence for an ${A}^{\ensuremath{'}}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}$ reaction, and set an upper limit of ${\ensuremath{\alpha}}^{\ensuremath{'}}/\ensuremath{\alpha}\ensuremath{\simeq}{10}^{\ensuremath{-}6}$. Our findings demonstrate that fixed-target searches can explore a new, wide, and important range of masses and couplings for sub-GeV forces.
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