Soft tissue elasticity has been a subject of interest in biomedical applications as an aid to medical diagnosis since the dawn of medicine. More recently, this has led to the concept of elastography with the aim of imaging the spatial distribution of tissue elasticity. Interpreting elastography images requires reliable information pertaining to elastic properties of normal and pathological tissues. Such information is either very limited or not available in the literature. Elastic modulus measurement techniques developed for soft tissues generally require tissue excision to prepare samples for testing. While this may be done with normal tissues, tumour tissue excision is generally not permissible because tumour pathological assessment requires that the tumour be kept intact. To address this limitation, we developed a system to measure the Young's modulus of tumour specimens. The technique consists of indenting the tumour specimen while measuring indentation force and displacements. To obtain the Young's modulus from the measured force-displacement slope, we developed an iterative inversion technique that uses a finite element model of the piecewise homogeneous tissue slice in each iteration. Preliminary elasticity measurement results of various breast tumours are presented and discussed. These results indicate that the proposed method is robust and highly accurate. Furthermore, they indicate that a benign lesion and malignant tumours are roughly five times and ten times stiffer than normal breast tissues respectively.