Phase contrast x-ray imaging is a powerful technique for the detection of low-contrast details in weakly absorbing objects. This method is of possible relevance in the field of diagnostic radiology. In fact, imaging low-contrast details within soft tissue does not give satisfactory results in conventional x-ray absorption radiology, mammography being a typical example. Nevertheless, up to now all applications of the phase contrast technique, carried out on thin samples, have required radiation doses substantially higher than those delivered in conventional radiological examinations. To demonstrate the applicability of the method to mammography we produced phase contrast images of objects a few centimetres thick while delivering radiation doses lower than or comparable to doses needed in standard mammographic examinations (typically similar to 1 mGy mean glandular dose (MGD)). We show images of a custom mammographic phantom and of two specimens of human breast tissue obtained at the SYRMEP bending magnet beamline at Elettra, the Trieste synchrotron radiation facility. The introduction of an intensifier screen enabled us to obtain phase contrast images of these thick samples with radiation doses comparable to those used in mammography. Low absorbing details such as 50 mu m thick nylon wires or thin calcium deposits (similar to 50 mu m) within breast tissue, invisible with conventional techniques, are detected by means of the proposed method. We also find that the use of a bending magnet radiation source relaxes the previously reported requirements on source size for phase contrast imaging. Finally, the consistency of the results has been checked by theoretical simulations carried out for the purposes of this experiment.