A quantitative characterization of the soft tissues composing the human breast is achieved by means of a monochromatic CT phase-contrast imaging system, through accurate measurements of their attenuation coefficients within the energy range of interest for breast CT clinical examinations. Quantitative measurements of linear attenuation coefficients are performed on tomographic reconstructions of surgical samples, using monochromatic x-ray beams from a synchrotron source and a free space propagation setup. An online calibration is performed on the obtained reconstructions, in order to reassess the validity of the standard calibration procedure of the CT scanner. Three types of healthy tissues (adipose, glandular, and skin) and malignant tumors, when present, are considered from each sample. The measured attenuation coefficients are in very good agreement with the outcomes of similar studies available in the literature, although they span an energy range that was mostly neglected in the previous studies. No globally significant differences are observed between healthy and malignant dense tissues, although the number of considered samples does not appear sufficient to address the issue of a quantitative differentiation of tumors. The study assesses the viability of the proposed methodology for the measurement of linear attenuation coefficients, and provides a denser sampling of attenuation data in the energy range useful to breast CT.