Cosmic ray intensity during the May 1959 magnetic storm is analysed, using neutron monitor and cubic telescope data from Mina Aguilar, Buenos Aires, Ushuaia and Ellsworth (Antarctica). The primary variation spectrum is estimated according to a method outlined in a recent paper (1). The intensity behaviour during recovery is analysed in detail. Following results are obtained. The primary variation spectrum, acting until the 18th of May, has an approximate form given byδD/D==δk(t)·E−0.5, valid up to very high energies. Around the 18th, this spectrum changes its shape at low energies (<2 GeV), in a similar way as it occured during the July 1959 storm recovery. This change may be interpreted as an additional removal of low energy particles, which lasted at least until the end of the month. On May 14, intensity variation is quite peculiar, in spite of being this a quiet day from the solar and geomagnetic point of view. A world-wide decrease occurs, with a superposed, 10 h lasting increase, which is particularly high at Ellsworth (9% in the neutron monitor, 2% for the cubical telescope). It is shown that Ellsworth was located in a 04 impact zone for a simultaneous solar event. On the 24th, a world-wide, small Forbush decrease occurred, associated with a magnetic storm and a decrease in the horizontal component of the magnetic field. This Forbush decrease, which shows a strong longitude dependence, corresponds to a primary variation spectrum which has the same shape as that responsible for the general recovery acting at the time,i.e., after the low energy change on the 18th.