This paper introduces a new technique for reconstruction of biomedical ultrasound images from simulated compressive measurements, based on modeling data with stable distributions. The proposed algorithm exploits two types of prior information: on one hand, our proposed approach is based on the observation that ultrasound RF echoes are best characterized statistically by 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 form the basis of an ℓp minimization approach that employs the iteratively reweighted least squares (IRLS) algorithm, but in which the parameter p is judiciously chosen, by relating it to the characteristic exponent of the underlying alpha-stable distributed data. We demonstrate, through Monte Carlo simulations, that the optimal value of the parameter p is just below that of the characteristic exponent α, which we estimate from the data. Our reconstruction results show that the proposed algorithm outperforms previously proposed reconstruction techniques, both visually and in terms of two objective evaluation measures.