Summary: Solving inverse problems is a fundamental component of science, engineering and mathematics. With the advent of deep learning, deep neural networks have significant potential to outperform existing state-of-the-art, model-based methods for solving inverse problems. However, it is known that current data-driven approaches face several key issues, notably hallucinations, instabilities and unpredictable generalization, with potential impact in critical tasks such as medical imaging. This raises the key question of whether or not one can construct deep neural networks for inverse problems with explicit stability and accuracy guarantees. In this work, we present a novel construction of accurate, stable and efficient neural networks for inverse problems with general analysis-sparse models, termed NESTANets. To construct the network, we first unroll NESTA, an accelerated first-order method for convex optimization. The slow convergence of this method leads to deep networks with low efficiency. Therefore, to obtain shallow, and consequently more efficient, networks we combine NESTA with a novel restart scheme. We then use compressed sensing techniques to demonstrate accuracy and stability. We showcase this approach in the case of Fourier imaging, and verify its stability and performance via a series of numerical experiments. The key impact of this work is demonstrating the construction of efficient neural networks based on unrolling with guaranteed stability and accuracy.