||When a type-I superconductor with a surface nucleation field Hc3(T)>Hc(T) (thermodynamic critical field) is thermally cycled in an axially applied magnetic field H0 between the temperatures T(Hc3) and about T(Hc), experiments show that the magnetization changes reversibly. The latter is diamagnetic near T(Hc3) but can be paramagnetic just above T(Hc). This behavior is explained by assuming that the fluxoid quantum number b is fixed at the transition from the normal to the superconducting state and retained at lower temperatures. The value of b is determined almost entirely by the flux at the transition which is enclosed by a contour located at a distance ξ1.7 from the surface inside the cylinder (ξ is the coherence length). The temperature variation of the order parameter f at the surface of the cylinder, the magnetization m, and the temperature at which m=0 for f≠0 are calculated for R>>ξ. Conservation of the fluxoid quantum number, while T is varied causes the two opposing surface currents to become imbalanced. This is the source of the observed para- and diamagnetism.