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This paper considers the detection-estimation problem for multiple uncorrelated plane waves impinging upon a so-called "partially augmentable" antenna array (whose difference set of intersensor spacings is incomplete). When the number of sources is not less than the number of contiguous (noninterrupted) covariance lags, detection-estimation involves the maximum-likelihood (ML) completion-estimation of some partially specified augmented Toeplitz covariance matrix. "Part I" in this series of papers (Abramovich et al. 2001) introduced and discussed a method for locally optimal Toeplitz covariance matrix estimation for "fully augmentable" arrays (that give rise to fully specified matrices). Here, this method is developed into locally optimal ML completion-estimation of a partially specified matrix. For identifiable scenarios, our completion technique yields an ideal restoration of the true covariance matrix when the specified covariance lags are exact. In the stochastic case, using the sample covariance matrix as a sufficient statistic, simulation results demonstrate a high detection-estimation performance.