Loss angle measurements in ultralow mechanical loss materials is normally affected by a large systematic error due to the excess losses introduced by the suspension system used to hold the samples. Crystals such as sapphire and silicon or amorphous materials such as fused silica can have loss angles in the range of 10(-10)-10(-7); such materials are of extreme interest in the detection of small displacements as it is required in quantum measurements, frequency stabilization, Micro Electro-Mechanical Systems (MEMS), and gravitational wave research. In the system proposed here the sample is suspended in equilibrium on top of a sphere, touching on one of the nodal points of vibration. The advantage of this system, as compared to others used so far, is twofold: (i) one surface only of the sample is touched and the contact surface is minimized because of the absence of applied forces; (ii) some relevant parameters of the suspension can be measured and eventually varied, giving the experimentalist the possibility to identify whether the measured loss is limited by the suspension system in use or it is an intrinsic property of the sample under investigation. The measurements of a 75 mm diameter and 3 mm thickness disk of Suprasil 311 gave a loss angle phi of 5x10(-8).