The present invention relates to a composition in the form of an oil-in-water emulsion, comprising, in a cosmetically acceptable medium: A) a continuous aqueous phase and B) an oily phase dispersed in the said aqueous phase and comprising at least one hydrocarbon-based oil; C) at least one mixture consisting of: i) at least one nonionic surfactant containing a saturated linear hydrocarbon-based chain comprising at least 16 carbon atoms, the said surfactant being other than a fatty alcohol, and ii) at least one fatty alcohol in pure form containing at least 16 carbon atoms or a mixture consisting exclusively of fatty alcohols containing at least 16 carbon atoms; D) at least one wax with a melting point of greater than 45° C. comprising one or more C40-C70 ester compounds and not comprising any C20-C39 ester compounds, in an amount from 1 to 10% by weight relative to the total weight of the composition and E) at least one water-soluble polysaccharide; the said composition having a viscosity, measured at 25° C. with a Rheomat RM 180 viscometer at 200 s−1 at room temperature with a No. 3 or No. 4 spindle after 10 minutes, ranging from 1500 mPa·s to 12 000 mPa·s and preferably from 2000 mPa·s to 8000 mPa·s. The present invention also relates to a cosmetic process for treating and/or caring for human keratin materials, characterized in that it consists in applying to the surface of the keratin material a composition as defined previously. The present invention also relates to a cosmetic process for treating human perspiration and/or perspiration-related body odour, which consists in applying to the surface of a human keratin material a composition as defined previously comprising at least one deodorant active agent and/or one antiperspirant active agent.