Abstract: We present the results of a missing-energy search for light dark matter which has a new interaction with ordinary matter transmitted by a vector boson, called dark photon A'. For the first time, this search is performed with a positron beam by using the significantly enhanced production of A' in the resonant annihilation of positrons with atomic electrons of the target nuclei, followed by the invisible decay of A' into dark matter. No events were found in the signal region with (10.1 +/- 0.1) x 109 positrons on target with 100 GeV energy. This allowed us to set new exclusion limits that, relative to the collected statistics, prove the power of this experimental technique. This measurement is a crucial first step toward a future exploration program with positron beams, whose estimated sensitivity is here presented.