The invention concerns a catalyst comprising a porous organic or inorganic support and at least one active phase comprising at least one acid selected from acids with formula R--SO.sub.3 H, where R is fluorine or an alkyl group or a fluorinated alkyl group, and at least one weakly basic aprotic organic solvent, the support having been impregnated with the active phase, the catalyst being such that it is essentially constituted by particles with an average diameter in the range 0.1 .mu.m to 150 .mu.m, and being such that before impregnation with the active phase, the support has a total pore volume in the range 0.5 cm.sup.3 /g to 6 cm.sup.3 /g. The invention also concerns the use of the catalyst for aliphatic alkylation.