The invention relates to a filling level measuring device (10) for measuring the filling level in a container (2) through the wall (9) thereof by means of ultrasound, having an ultrasonic measuring head (12), a controller (20), and a fastening device (24) by means of which the filling level measuring device (10) can be fastened to the container (2) such that the ultrasonic measuring head (12) is pressed against the wall (9) of the container (2). The invention further relates to a method of operating such a filling level measuring device (10), wherein a sampling rate is used which is situation-dependent. The invention finally relates to an assembly made up of such a filling level measuring device and at least one spacer (50) which can be attached to the lower edge of a container (2) to be provided with the filling level measuring device (10).