A catalyst comprising a porous organic or mineral support, preferably silica, and an acidic phase containing B(OSO.sub.2 CF.sub.3).sub.3 and at least one acid selected from the group formed by sulphuric acid (H.sub.2 SO.sub.4) and trifluoromethane sulphonic acid (CF.sub.3 SO.sub.3 H), said support having been impregnated with said acidic phase and said support prior to impregnation having a total pore volume of 0.1-0.6 cm.sup.3 /g, said catalyst being essentially constituted by particles with an average diameter of between 0.1 and 30 .mu.m, said catalyst being characterized in that said acidic phase contains:between 0.1 and 70% by weight of B(OSO.sub.2 CF.sub.3).sub.3 ;between 0 and 90% by weight of H.sub.2 SO.sub.4 ;between 0 and 90% by weight of CF.sub.3 SO.sub.3 H.is suitable for the catalytic alkylation of isobutane and/or isopentane in the presence of at least one olefin containing 2 to 6 carbon atoms per molecule, preferably 3 to 6 carbon atoms per molecule.