Superfluids are distinguished from normal fluids by their peculiar response(1) to rotation: circulating flow in superfluid helium(2,3), a strongly coupled Bose liquid, can appear only as quantized vortices(4-6). The newly created Bose-Einstein condensates(7,9)- clouds of millions of ultracold, weakly interacting alkali-metal atoms that occupy a single quantum state-offer the possibility of investigating superfluidity in the weak-coupling regime, An outstanding question is whether Bose-Einstein condensates exhibit a mesoscopic quantum analogue of the macroscopic vortices in superfluids, and what its experimental signature would be, Here we report calculations of the low-energy states of a rotating, weakly interacting Bose gas. We find a succession of transitions between stable vortex patterns of differing symmetries that are in general qualitative agreement with observations(5) of rotating superfluid helium, a strong-coupling superfluid, Counterintuitively, the angular momentum per particle is not quantized. Some angular momenta are forbidden, corresponding to asymmetrical unstable states that provide a physical mechanism for the entry of vorticity into the condensate.