An experiment is described in which the picosecond temporal structure of a Cerenkov radiation source has been measured using an optical streak camera. Electrons in the Stanford Linear Accelerator Center (SLAC) three kilometer linear accelerator, bunched by the 2856 MHz acceleration fields, are used to generate Cerenkov radiation in a helium-filled gas cell. We show that it is possible to construct a Cerenkov cell which introduces subpicosecond time spread and allows for nondestructive monitoring of the electron beam.