The properties of a superconductor are expected to change radically when its size becomes comparable to that of the Cooper pairs, the quasiparticles responsible for superconductivity. The effect of such confinement is well understood for the case of the suppression of superconductivity by magnetic fields (which gives rise to so-called Little-Parks oscillations of the phase boundary)(1-4). But little is known about what happens in small superconductors in the zero-resistance state, which cannot be probed by resistance measurements. Here we apply a new technique of ballistic Hall magnetometry(5) to study the magnetization of individual superconducting discs of diameters down to 100 nm. The superconducting state of these discs is found to be qualitatively different from both macroscopic and microscopic(6) superconductors, with numerous phase transitions whose character changes rapidly with size and temperature. This exotic behaviour is due to size quantization of the Cooper-pair motion and resulting transitions between discrete states of the superconducting Bose condensate in a magnetic field.