The formation of a Bose-Einstein condensate of a dilute atomic gas has been studied in situ with a nondestructive, time-resolved imaging technique. Sodium atoms were evaporatively cooled close to the onset of Bose-Einstein condensation and then suddenly quenched to below the transition temperature. The subsequent equilibration and condensate formation showed a slow onset distinctly different from simple relaxation. This behavior provided evidence for the process of bosonic stimulation, or coherent matter-wave amplification, crucial to the concept of an atom laser.