A high torsional stiffness low axial deflection stiffness flexible drive shaft apparatus is provided for driving a magnetic tape cassette and includes a drive shaft having opposite ends, a device for engaging the magnetic tape cassette and a pair of flexible diaphragms. A device is connected to one of the drive shaft ends, one of the flexible diaphragms, and is connectable to a motor for transmitting torsional force from the motor to the drive shaft via the flexible diaphragm. Another device is connected to the other end of the drive shaft, the other flexible diaphragm and the tape cassette engaging device for transmitting torsional force from the drive shaft to the engaging device via the other flexible diaphragm. Because the diaphragms are too delicate to handle thrust loads and because the torsional stiffness of the diaphragm degrades rapidly as its center is displaced axially with respect to the outer driving points, a device is operably connected to each torsional force transmitting device for transmitting thrust loads exerted along the axes of rotation of the drive apparatus. With this arrangement the drive apparatus can easily adapt for misalignment of the drive apparatus with the magnetic tape cassette to deliver a reliable rotating force.