A brake device of an elevator car with a brake unit that is mounted on the elevator car. The brake drive includes a flexible tension member that passes around two tension-members reversing-sheaves. One end of the tension members extends parallel to the travel path of the elevator car and is coupled to an actuating element of the brake unit, and thereby also to the elevator car, so that in normal operation the tension member moves synchronously with the elevator car. A blocking device, at least on overspeed of the elevator car, brakes the tension member, as a consequence of which the tension member, via the actuating element, activates the brake unit. A monitoring device detects failure of the coupling between the tension member and the elevator car.