Household appliance (2) for the supplying of cooled water or other beverage having a tank (4) communicating directly with a source (8) supplying the beverage at above atmospheric pressure; means (5) for controlling the outflow of the water or other beverage from the tank (4); a cooling assembly (9) for cooling the beverage inside the tank (4); and a control unit (10) for controlling the cooling assembly (9) to bring the beverage inside the tank (4) to, and maintain it at, a reference temperature below or equal to the freezing temperature (T 0 ) of the beverage; the household appliance (2) also having freezing control means (13) for assisting irregular, non-homogeneous crystallization of the beverage to obtain a semisolid frozen drink mixture.