  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 $4.3\times10^{10}$ 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.
