The growth of a tumour in a cylindrical duct with compliant walls is examined in order to model the early stages of ductal carcinoma in situ (DCIS) of the breast, the earliest known stage of breast cancer. A nutrient-limited growth model is formulated, in which cell movement is described by a Stokes flow constitutive relation. The interactions between the expansive forces created by tumour cell proliferation and the stresses that develop in the compliant basement membrane are studied using asymptotic and numerical methods. In this way we show how the duct wall deforms as the tumour grows and also how the progression of the tumour along the duct depends upon the stiffness of the wall. By varying key parameters we determine how treatment, protease production and the inclusion of the surrounding stroma affect the growth. Finally, we discuss the biological relevance of our results and suggest possible directions for future work.