  We present a search at Jefferson Laboratory for new forces mediated by
sub-GeV vector bosons with weak coupling $\alpha'$ to electrons. Such a
particle $A'$ can be produced in electron-nucleus fixed-target scattering and
then decay to an $e^+e^-$ 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 $A'\to e^+e^-$ reaction, and set an upper limit of
$\alpha'/\alpha \simeq 10^{-6}$. Our findings demonstrate that fixed-target
searches can explore a new, wide, and important range of masses and couplings
for sub-GeV forces.
