In this letter, a deep learning framework for direction of arrival (DOA) estimation is developed. We first show that the columns of the array covariance matrix can be formulated as under-sampled noisy linear measurements of the spatial spectrum. Then, a deep convolution network (DCN) that learns the inverse transformation from large training dataset is introduced. In contrast to traditional sparsity-inducing methods with computationally complex iterations, the proposed DCN-based framework could efficiently obtain DOA estimates in near real time. Moreover, the utilization of the sparsity prior improves DOA estimation performance compared to existing deep learning based methods. Simulation results have demonstrated the superiority of the proposed method in both DOA estimation precision and computation efficiency especially when SNR is low.