A search is performed for a new sub-GeV vector boson ($A$) mediated production of Dark Matter ($chi$) in the fixed-target experiment, NA64, at the CERN SPS. The $A$, called dark photon, could be generated in the reaction $ e^- Z to e^- Z A$ of 100 GeV electrons dumped against an active target which is followed by the prompt invisible decay $A to chi overline{chi}$. The experimental signature of this process would be an event with an isolated electron and large missing energy in the detector. From the analysis of the data sample collected in 2016 corresponding to $4.3times10^{10}$ electrons on target no evidence of such a process has been found. New stringent constraints on the $A$ mixing strength with photons, $10^{-5}lesssim epsilon lesssim 10^{-2}$, for the $A$ mass range $m_{A} lesssim 1$ GeV are derived. For models considering scalar and fermionic thermal Dark Matter interacting with the visible sector through the vector portal the 90% C.L. limits $10^{-11}lesssim y lesssim 10^{-6}$ on the dark-matter parameter $y = epsilon^2 alpha_D (frac{m_chi}{m_{A}})^4 $ are obtained for the dark coupling constant $alpha_D = 0.5$ and dark-matter masses $0.001 lesssim m_chi lesssim 0.5 $ GeV. The lower limits $alpha_D gtrsim 10^{-3} $ for pseudo-Dirac Dark Matter in the mass region $m_chi lesssim 0.05 $ GeV are more stringent than the corresponding bounds from beam dump experiments. The results are obtained by using tree level, exact calculations of the $A$ production cross-sections, which turn out to be significantly smaller compared to the one obtained in the Weizs{a}cker-Williams approximation for the mass region $m_{A} gtrsim 0.1$ GeV.
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