  The process of the triplet production on a free electron, $\gamma e^-\to
e^+e^-e^-$, has been investigated as a reaction where a dark photon, $A'$, is
produced as a virtual state with subsequent decay into a $e^+e^-$-pair. This
effect arises due to the so-called kinetic mixing and is characterized by the
small parameter $\epsilon$ describing the coupling strength relative to the
electric charge e.
  The search of $A'$ in this process has advantage because the background to
the $A'$ signal is pure QED. This QED background is described by eight Feynman
diagrams taking into account the identity of final electrons. As concern $A'$,
we leave its contribution in Compton-like diagrams only since, in this case,
the virtual dark photon is time-like and its propagator has the Breit-Wigner
form. So, near the resonance $A'$ can manifest itself. The contribution of $A'$
in Borsellino diagrams is negligible since, in this case, the virtual dark
photon is space-like, the $A'$ propagator does not peak and effect is
proportional at least to $\epsilon^2$. We calculate the distributions over the
invariant masses of the both produced $e^+e^-$ pairs and search for the
kinematical region where the Compton-like diagrams contribution is not
suppressed as compared with the Borsellino one. We estimate what value of the
parameter $\epsilon$, as a function of the dark photon mass, can be obtained at
given number of the measured events.
