A new design of a detector plane of sub-millimetre thickness for an electromagnetic sampling calorimeter is presented. It is intended to be used in the luminometers LumiCal and BeamCal in future linear $e^+e^-$ collider experiments. The detector planes were produced utilising novel connectivity scheme technologies. They were installed in a compact prototype of the calorimeter and tested at DESY with an electron beam of energy 1-5 GeV. The performance of a prototype of a compact LumiCal comprising eight detector planes was studied. The effective Moli`ere radius at 5 GeV was determined to be (8.1 +/- 0.1 (stat) +/- 0.3 (syst)) mm, a value well reproduced by the Monte Carlo (MC) simulation (8.4 +/- 0.1) mm. The dependence of the effective Moli`ere radius on the electron energy in the range 1-5 GeV was also studied. Good agreement was obtained between data and MC simulation.
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