We show that a distribution of micrometer-sized calcifications in the human breast which are not visible in clinical x-ray mammography at diagnostic dose levels can produce a significant dark-field signal in a grating-based x-ray phase-contrast imaging setup with a tungsten anode x-ray tube operated at 40 kVp. A breast specimen with invasive ductal carcinoma was investigated immediately after surgery by Talbot-Lau x-ray interferometry with a design energy of 25 keV. The sample contained two tumors which were visible in ultrasound and contrast-agent enhanced MRI but invisible in clinical x-ray mammography, in specimen radiography and in the attenuation images obtained with the Talbot-Lau interferometer. One of the tumors produced significant dark-field contrast with an exposure of 0.85 mGy air-kerma. Staining of histological slices revealed sparsely distributed grains of calcium phosphate with sizes varying between 1 and 40 μm in the region of this tumor. By combining the histological investigations with an x-ray wave-field simulation we demonstrate that a corresponding distribution of grains of calcium phosphate in the form of hydroxylapatite has the ability to produce a dark-field signal which would-to a substantial degree-explain the measured dark-field image. Thus we have found the appearance of new information (compared to attenuation and differential phase images) in the dark-field image. The second tumor in the same sample did not contain a significant fraction of these very fine calcification grains and was invisible in the dark-field image. We conclude that some tumors which are invisible in x-ray absorption mammography might be detected in the x-ray dark-field image at tolerable dose levels.