  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.
