In this paper, we introduce a new sparsity-promoting prior, namely, the "normal product" prior, and develop an efficient algorithm for sparse signal recovery under the Bayesian framework. The normal product distribution is the distribution of a product of two normally distributed variables with zero means and possibly different variances. Like other sparsity-encouraging distributions such as the Student's $t$-distribution, the normal product distribution has a sharp peak at origin, which makes it a suitable prior to encourage sparse solutions. A two-stage normal product-based hierarchical model is proposed. We resort to the variational Bayesian (VB) method to perform the inference. Simulations are conducted to illustrate the effectiveness of our proposed algorithm as compared with other state-of-the-art compressed sensing algorithms.