A process for purifying a hydrocarbon feed comprising impurities, in which the following steps are carried out simultaneously: a) treating the hydrocarbon feed in the liquid phase in a first adsorption unit comprising a first and a second adsorption column (1, 2) respectively filled with a first and second adsorbent solid, the first and second adsorption columns (1,2) operating in parallel and alternately in adsorption mode and in regeneration mode, said hydrocarbon feed being introduced into the first adsorption column and brought into contact with the first adsorbent solid, and a hydrocarbon effluent which is depleted in impurities is taken off at the outlet from the first adsorption column; b) treating a secondary liquid hydrocarbon feed which is constituted either by a fraction of the hydrocarbon feed or by a fraction of the hydrocarbon effluent which is depleted in impurities in a treatment unit (3, 4, 22, 24) comprising a third and fourth adsorption column (3, 4) operating in parallel and alternately in adsorption mode and in regeneration mode, said columns respectively comprising a third and fourth adsorbent solid, and a treated secondary liquid hydrocarbon feed is taken off from said treatment unit; c) heating the treated secondary liquid hydrocarbon feed obtained from step b); d) regenerating the second adsorbent solid of the second adsorption column with the secondary hydrocarbon feed heated in step c) by bringing said feed into contact with the second adsorbent solid in order to desorb the impurities from the second adsorbent solid and produce an effluent which is charged with impurities, in which the secondary liquid hydrocarbon feed is brought into contact in the third adsorption column with the third adsorbent solid so as to produce the treated secondary liquid hydrocarbon feed and in which the effluent charged with impurities obtained from the second adsorption column is sent to the fourth adsorption column so as to regenerate the fourth adsorbent solid and evacuate an effluent charged with impurities. a) treating the hydrocarbon feed in the liquid phase in a first adsorption unit comprising a first and a second adsorption column (1, 2) respectively filled with a first and second adsorbent solid, the first and second adsorption columns (1,2) operating in parallel and alternately in adsorption mode and in regeneration mode, said hydrocarbon feed being introduced into the first adsorption column and brought into contact with the first adsorbent solid, and a hydrocarbon effluent which is depleted in impurities is taken off at the outlet from the first adsorption column; b) treating a secondary liquid hydrocarbon feed which is constituted either by a fraction of the hydrocarbon feed or by a fraction of the hydrocarbon effluent which is depleted in impurities in a treatment unit (3, 4, 22, 24) comprising a third and fourth adsorption column (3, 4) operating in parallel and alternately in adsorption mode and in regeneration mode, said columns respectively comprising a third and fourth adsorbent solid, and a treated secondary liquid hydrocarbon feed is taken off from said treatment unit; c) heating the treated secondary liquid hydrocarbon feed obtained from step b); d) regenerating the second adsorbent solid of the second adsorption column with the secondary hydrocarbon feed heated in step c) by bringing said feed into contact with the second adsorbent solid in order to desorb the impurities from the second adsorbent solid and produce an effluent which is charged with impurities,