The invention relates to an exchanger, advantageously a plate exchanger, comprising counter-current parallel exchange sections (8,9). According to the invention, the exchange section for the water (8) has an essentially constant cross-section, whereas the exchange section (9) for the cryogen has a reduced inlet region (9a) enabling the possible formation of ice (10) in the water passage (8) to be concentrated on a small volume, without blocking said water passage. The invention can especially be applied to the supply of gaseous hydrogen stored in a liquid form to an on-board power generating device in a vehicle.