Process for cryogenic cooling of a first fluid available at about ambient temperature, by countercurrent heat exchange with a second fluid available at a cryogenic temperature below 20° C., which comprises alternately circulating the first fluid in a heat exchange apparatus (3) that comprises a vessel containing heat exchange material and that has a radially external passageway and a radially internal passageway, in an overall centrifugal manner relative to a general central axis (X-X) of this apparatus, from said radially external passageway (17) of the apparatus through said heat exchange material and out said radially external passageway, and circulating the second fluid in a generally centrifugal manner relative to said central axis, from said radially internal passageway (18) through said heat exchange material and out said radially external passageway of the apparatus.