A search for sub-GeV dark matter production mediated by a new vector boson A^{'}, called a dark photon, is performed by the NA64 experiment in missing energy events from 100 GeV electron interactions in an active beam dump at the CERN SPS. From the analysis of the data collected in the years 2016, 2017, and 2018 with 2.84×10^{11} electrons on target no evidence of such a process has been found. The most stringent constraints on the A^{'} mixing strength with photons and the parameter space for the scalar and fermionic dark matter in the mass range ≲0.2 GeV are derived, thus demonstrating the power of the active beam dump approach for the dark matter search.