This paper introduces a new technique for compressive sampling reconstruction of biomedical ultrasound images that exploits two types of prior information. On the one hand, our proposed approach is based on the observation that ultrasound RF echoes are best characterised statistically using alpha-stable distributions. On the other hand, through knowledge of the acquisition process, the support of the RF echoes in the Fourier domain can be easily inferred. Together, these two facts inform an iteratively reweighted least squares (IRLS) algorithm, which is shown to outperform previously proposed reconstruction techniques, both visually and in terms of two objective evaluation measures.