The ZEUS detector at HERA has been supplemented with a presampler detector in front of the forward and rear calorimeters. It consists of a segmented scintillator array read out with wavelength-shifting fibers. We discuss its design, construction and performance. Test beam data obtained with a prototype presampler and the ZEUS prototype calorimeter demonstrate the main function of this detector, i.e. the correction for the energy lost by an electron interacting in inactive material in front of the calorimeter.