A superconductor placed in a magnetic field and cooled down through the transition temperature expels magnetic flux. This phenomenon, known as the Meissner effect, is arguably the most essential property of superconductors and implies zero resistivity. Surprisingly, several recent experiments have shown that some superconducting samples(1-7) may attract magnetic field-the so-called paramagnetic Meissner effect. The scarce, if not controversial, experimental evidence for this effect makes it difficult to identify the origin of this enigmatic phenomenon, although a large number of possible explanations have been advanced(8-16). Here we report observations of the paramagnetic Meissner effect with a resolution better than one quantum of magnetic flux. The paramagnetic Meissner effect is found to be an oscillating function of the magnetic field (due to flux quantization) and replaces the normal Meissner effect only above a certain field when several nux quanta are frozen inside a superconductor. The paramagnetic state is found to be metastable and the Meissner state can be restored by external noise. We conclude that the paramagnetic Meissner effect is related to the surface superconductivity and, therefore, represents a general property of superconductors : on decreasing temperature, the flux captured at the third (surface) critical field inside the superconducting sheath compresses into a smaller volume, allowing extra flux to penetrate at the surface.