A system for the preparation of beverages, in particular milk-based beverages, such as cappuccino, chocolate and the like, comprising a coffee-making machine provided with an outlet fluid socket connected or able to be connected to an internal steam source of the machine, and an accessory apparatus manufactured as a separate apparatus able to be connected to the coffee-making machine; the accessory apparatus comprising a container, a removable cover, suitable for closing the container, a rotatable stirrer which extends inside the container and is provided with an end impeller, the cover being designed to support the stirrer rotationally, an electric motor, capable of driving the stirrer rotationally via a coupling joint (9, 10, 23, 26), and a hot steam dispensing wand, also extending inside the container and connectable to the steam source of said machine via a fluid connection union associated with the cover; the system being characterized in that said electric motor is contained in a housing which is separate and distinct from the cover and is connected or able to be connected to power supply terminals (49, 50) of the coffee-making machine; the inlet fluid connector of the cover is able to be directly connected to the steam dispensing fluid union or socket of the coffee-making machine; the aforementioned coupling joint comprising first and second coupling means (9, 10; 23, 26) which are separate and distinct from each other and are connected to the shaft of the motor and to the end of the stirrer opposite the impeller and close to the cover, respectively; the housing and the cover being able to be coupled together in an operating configuration in which said first and second coupling means (9, 10; 23, 26) allow transmission of the rotation from the motor to the stirrer; and in that the coffee-making machine is provided with means (55, 56; ECU) for detecting whether the assembly formed by the container and the cover of the apparatus is connected to said electric motor, said detection means (55, 56) comprising means (ECU; 55) for detecting the amount of power used by said electric motor during a period of preliminary activation thereof and comparing the amount of power thus detected with at least one predetermined threshold value. a coffee-making machine provided with an outlet fluid socket connected or able to be connected to an internal steam source of the machine, and an accessory apparatus manufactured as a separate apparatus able to be connected to the coffee-making machine; the accessory apparatus comprising a container, a removable cover, suitable for closing the container, a rotatable stirrer which extends inside the container and is provided with an end impeller, the cover being designed to support the stirrer rotationally, an electric motor, capable of driving the stirrer rotationally via a coupling joint (9, 10, 23, 26), and a hot steam dispensing wand, also extending inside the container and connectable to the steam source of said machine via a fluid connection union associated with the cover; said electric motor is contained in a housing which is separate and distinct from the cover and is connected or able to be connected to power supply terminals (49, 50) of the coffee-making machine; the inlet fluid connector of the cover is able to be directly connected to the steam dispensing fluid union or socket of the coffee-making machine; the aforementioned coupling joint comprising first and second coupling means (9, 10; 23, 26) which are separate and distinct from each other and are connected to the shaft of the motor and to the end of the stirrer opposite the impeller and close to the cover, respectively; the housing and the cover being able to be coupled together in an operating configuration in which said first and second coupling means (9, 10; 23, 26) allow transmission of the rotation from the motor to the stirrer; and in that the coffee-making machine is provided with means (55, 56; ECU) for detecting whether the assembly formed by the container and the cover of the apparatus is connected to said electric motor, said detection means (55, 56) comprising means (ECU; 55) for detecting the amount of power used by said electric motor during a period of preliminary activation thereof and comparing the amount of power thus detected with at least one predetermined threshold value.