A new device for measuring the rheological properties of soft biological tissues is presented. The mechanical response is characterized for harmonic shear deformations at high frequencies (up to 10 kHz) and small strains (up to 0.2%). Experiments are performed using a cylindrical rod excited to torsional resonance. One extremity of the rod is in contact with the soft tissue and adherence is ensured by vacuum clamping. The damping characteristics and the resonance frequency of the vibrating system are inferred from the control variables of a phase stabilization loop. Due to the contact with the soft tissue, and depending on the rheological properties of the tissue, changes occur in the Q-factor and in the resonance frequency of the system. The shear modulus of the soft tissue is determined from the experimental results with an analytical model. The reliability of the proposed technique is evaluated through repeatability tests and comparative measurements with synthetic materials. The results of measurements on bovine organs demonstrate the suitability of the experimental procedure for the characterization of biological tissues and provide some insight in their rheological properties at frequencies in the range 1-10 kHz.