We have observed the persistent flow of Bose-condensed atoms in a toroidal trap. The flow persists without decay for up to 10 s, limited only by experimental factors such as drift and trap lifetime. The quantized rotation was initiated by transferring one unit of the orbital angular momentum from Laguerre-Gaussian photons to each atom. Stable flow was only possible when the trap was multiply connected, and was observed with a Bose-Einstein condensate fraction as small as 20 . We also created flow with two units of angular momentum and observed its splitting into two singly charged vortices when the trap geometry was changed from multiply to simply connected.