Several models of dark matter suggest the existence of dark sectors consisting of SU(3)_C x SU(2)_L x U(1)_Y singlet fields. These sectors of particles do not interact with the ordinary matter directly but could couple to it via gravity. In addition to gravity, there might be another very weak interaction between the ordinary and dark matter mediated by U(1) gauge bosons A (dark photons) mixing with our photons. In a class of models the corresponding dark gauge bosons could be light and have the $gamma$-A coupling strength laying in the experimentally accessible and theoretically interesting region. If such A mediators exist, their di-electron decays A -> e+e- could be searched for in a light-shining-through-a-wall experiment looking for an excess of events with the two-shower signature generated by a single high energy electron in the detector. A proposal to perform such an experiment aiming to probe the still unexplored area of the mixing strength 10^-5 < $epsilon$ < 10^-3 and masses M_A < 100 MeV by using 10-300 GeV electron beams from the CERN SPS is presented. The experiment can provide complementary coverage of the parameter space, which is intended to be probed by other searches. It has also a capability for a sensitive search for As decaying invisibly to dark-sector particles, such as dark matter, which could cover a significant part of the still allowed parameter space. The full running time of the proposed measurements is requested to be up to several months, and it could be taken at different SPS secondary beams.
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