This paper deals with structured covariance matrix estimation in a robust statistical framework. Covariance matrices often exhibit a particular structure related to the application of interest and taking this structure into account increases estimation accuracy. Within the framework of robust estimation, the class of circular Complex Elliptically Symmetric (CES) distributions is particularly interesting to handle impulsive and spiky data. Normalized CES random vectors are known to share a common Complex Angular Elliptical distribution. In this context, we propose a Robust Covariance Matrix Estimation Technique (RCOMET) based on Tyler’s estimate and COMET criterion for convexly structured matrices. We prove that the proposed estimator is consistent and asymptotically efficient while computationally attractive. Numerical results support the theoretical analysis in a particular application for Hermitian Toeplitz structure.