This paper presents an efficient method for computing maximum likelihood (ML) direction of arrival (DOA) estimates assuming unknown sensor noise powers. The method combines efficient Alternate Projection (AP) procedures with Newton iterations. The efficiency of the method lies in the fact that all its intermediate steps have low complexity. The main contribution of this paper is the method's last step, in which a concentrated cost function is maximized in both the DOAs and noise powers in a few iterations through a Newton procedure. This step has low complexity because it employs closed-form expressions of the cost function's gradients and Hessians, which are presented in the paper. The method's total computational burden is of just a few mega-flops in typical cases. We present the method for the deterministic and stochastic ML estimators. An analysis of the deterministic ML cost function's gradient reveals an unexpected drawback of its associated estimator: if the noise powers are unknown, then it is either degenerate or inconsistent. The root-mean-square (RMS) error performance and computational burden of the method are assessed numerically.