A device for the aftertreatment of an exhaust gas of an internal combustion engine flowing through an exhaust line includes at least one dosing unit disposed in an opening of an exhaust pipe of the exhaust line and configured to add an exhaust-gas aftertreatment agent into the exhaust line. The dosing unit is enclosed by a cooling jacket, through which the dosing unit is fastened in the opening of the exhaust pipe. The cooling jacket is at least partially enclosed by a space or chamber delimited at least by the cooling jacket, the exhaust pipe, and a cover plate. The cover plate is connected to the exhaust pipe and spaced apart from the cooling jacket or the dosing unit by a gap. A motor vehicle having the device is also provided.