A maximum-likelihood estimator (MLE) for coherence is derived. The MLE should be unbiased, and there should be no unbiased estimator with smaller variance. An approximation to the MLE, valid for coherence estimates with a large number of degrees of freedom (DOF), is also derived. The two MLE algorithms and two estimators discussed in earlier literature are compared in a simulation consisting of 8192 calculations of coherence estimates, each with 8 DOF. The MLE calculations are consistently closer to the design coherences than the other estimators. The approximation to the MLE performs surprisingly well, even when the coherence estimates have only 8 DOF. >