Recent matrix completion based methods have not been able to properly model the Haplotype Assembly Problem (HAP) for noisy observations. To cope with such a case, in this letter we propose a new Minimum Error Correction (MEC) based matrix completion optimization problem over the manifold of rank-one matrices. The convergence of a specific iterative algorithm for solving this problem is proved. Simulation results illustrate that the proposed method not only outperforms some well-known matrix completion based methods, but also presents a more accurate result compared to a most recent MEC based algorithm for haplotype estimation.