Single ionization of He by 1 keV electron impact in the presence of an intense (I=4 x 10(12) W/cm(2)) laser field (lambda=1064 nm) has been explored in a kinematically complete experiment using a reaction microscope. Distinct differences in the singly to fully differential cross sections compared to the field-free situation are observed which cannot be explained by a first-order quantum calculation. Major features, such as the number of photons exchanged and the modification of the energy spectrum of emitted electrons, can be understood qualitatively within a simple classical model.