A novel matrix completion ESPRIT (MC-ESPRIT) algorithm is proposed to estimate the direction of arrival (DOA) with nonuniform linear arrays (NLA). By exploiting the matrix completion theory and the characters of Hankel matrix, the received data matrix of an NLA is tranformed into a two-fold Hankel matrix, which is a treatable for matrix completion. Then the decision variable can be reconstructed by the inexact augmented Lagrange multiplier method. This approach yields a completed data matrix, which is the same as the data matrix of uniform linear array (ULA). Thus the ESPRIT-type algorithm can be used to estimate the DOA. The MC-ESPRIT could resolve more signals than the MUSIC-type algorithms with NLA. Furthermore, the proposed algorithm does not need to divide the field of view of the array compared to the existing virtual interpolated array ESPRIT (VIA-ESPRIT). Simulation results confirm the effectiveness of MC-ESPRIT.