Using a focused laser beam we stir a Bose-Einstein condensate of ${}^{87}\mathrm{Rb}$ confined in a magnetic trap and observe the formation of a vortex for a stirring frequency exceeding a critical value. At larger rotation frequencies we produce states of the condensate for which up to four vortices are simultaneously present. We have also measured the lifetime of the single vortex state after turning off the stirring laser beam.