Abstract: The current most stringent constraints for the existence of sub-GeV dark matter coupling to Standard Model via a massive vector boson A' were set by the NA64 experiment for the mass region m(A') less than or similar to 250 MeV, by analyzing data from the interaction of 2.84 x 10(11) 100-GeV electrons with an active thick target and searching for missing-energy events. In this work, by including A' production via secondary positron annihilation with atomic electrons, we extend these limits in the 200-300 MeV region by almost an order of magnitude, touching for the first time the dark matter relic density constrained parameter combinations. Our new results demonstrate the power of the resonant annihilation process in missing energy dark-matter searches, paving the road to future dedicated e(+) beam efforts.