An experimental study of the mechanical response of frozen soft biological tissues to applied compressive stresses is presented. This study is related to the mechanical stresses that develop due to the contraction of frozen tissues in cryopreservation as well as in cryosurgical procedures. The main concept in this study is that the stresses associated with the constrained contraction of the frozen tissue, i.e., due to temperature variations within the frozen tissue, can be simulated by an external mechanical load which is applied to the frozen tissue while the tissue is maintained at a uniform temperature. A new apparatus for measuring compressive stresses and strains of frozen biological tissues in cryogenic temperature range is presented. A new technique for processing the fresh biological tissue into a cylindrical frozen sample for mechanical testing is introduced. Results of compression tests on rabbit liver, kidney, and brain are presented and are compared with available data from the literature on sea ice and single ice crystals. An unusual response of frozen biological tissues to compressive stress was observed: after the initial, roughly linear elastic portion there was a series of sudden stress drops at constant strain, each followed by a linear increase of stress with strain to the next drop. This phenomenon, which is attributed to the accumulation of micro cracks, broadly resembles plastic deformation, and thus provides some support for simple mechanical models invoked in theoretical studies.