The paramagnetic Meissner effect (PME), recently observed in high-T-c materials and also in Nb, can be successfully explained by the persistence of a giant vortex state with a fixed orbital quantum number L. This state is formed in superconductors in the field-cooled regime at the third critical field. The self-consistent numerical solution of the Ginzburg-Landau equations clearly shows that the compression of the flux trapped inside the giant vortex state can result in the PME. The PME is suppressed, and the normal diamagnetic response is recovered, by increasing the applied field. A possible definition of the irreversibility line, as a crossover between the giant vortex state and the Abrikosov flux line lattice, is discussed. The transition between the two quantum states (L = 0 and L = 1) has been used to calculate the field H0-->1(T), corresponding to the penetration of the first flux line into a cylindrical sample.