In addition to gravity, there might be another very weak interaction between
the ordinary and dark matter transmitted by U'(1) gauge bosons A' (dark
photons) mixing with our photons. If such A's exist, they could be searched for
in a light-shining-through-a-wall experiment with a high energy electron beam.
The electron energy absorption in a calorimeter (CAL1) is accompanied by the
emission of bremsstrahlung A's in the reaction eZ -> eZA' of electrons
scattering on nuclei due to the \gamma - A' mixing. A part of the primary beam
energy is deposited in the CAL1, while the rest of the energy is transmitted by
the A' through the "CAL1 wall" and deposited in another downstream calorimeter
CAL2 by the e+e- pair from the A'->e+e- decay in flight. Thus, the A's could be
observed by looking for an excess of events with the two-shower signature
generated by a single high energy electron in the CAL1 and CAL2. A proposal to
perform such an experiment 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
A's decaying invisibly to dark-sector particles, such as dark matter, which
could cover a significant part of the still allowed parameter space.Comment: 10 pages, 7 figures. Discussion on a sensitive search for the A'->
invisible decay added. This work is continued in arXiv:1312.3309. To appear
in PR