We have made an observation of 45 MeV electron beam bunches using the nondestructive electro-optical (EO) technique. The amplitude of the EO modulation was found to increase linearly with electron beam charge and decrease inversely with the optical beam path distance from the electron beam. The risetime of the signal was bandwidth limited by our detection system to ~ 70 ps. An EO signal due to ionization caused by the electrons traversing the EO crystal was also observed. The EO technique may be ideal for the measurement of bunch structure with femtosecond resolution of relativistic charged particle beam bunches.