In order to measure precisely and fast the luminosity in a future experiment at a linear collider like ILC or CLIC two compact and fine grained electromagnetic calorimeters are foreseen, LumiCal and BeamCal. LumiCal will provide a precise measurement of the integrated luminosity by counting Bhabha scattering events at small polar angles, while BeamCal is designed for bunch-by-bunch luminosity measurement and beam tuning when included in a fast feedback system using beam-strahlung. Both detectors will extend the detector coverage to small polar angles, potentially important for tagging high-energy electrons. The performance of a prototype of LumiCal, constructed as a sampling calorimeter with tungsten absorber plates of one X0 thickness and sensor planes with sub-mm thickness, was studied in an electron beam at DESY with momenta between one and 5 GeV. The shower position is measured with a precision of 440±20 microns, and the effective Molière radius is determined for electrons of 5 GeV to be 8.1±0.3mm.