Particles with electric charge q < 10^(-3)e and masses in the range 1--100 MeV/c^2 are not excluded by present experiments. An experiment uniquely suited to the production and detection of such millicharged particles has been carried out at SLAC. This experiment is sensitive to the infrequent excitation and ionization of matter expected from the passage of such a particle. Analysis of the data rules out a region of mass and charge, establishing, for example, a 95%-confidence upper limit on electric charge of 4.1X10^(-5)e for millicharged particles of mass 1 MeV/c^2 and 5.8X10^(-4)e for mass 100 MeV/c^2.