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This paper deals with the problem of blind separation of an instantaneous mixture of Gaussian autoregressive sources, without additive noise, by the exact maximum likelihood approach. The maximization of the likelihood function is divided, using relaxation, into two suboptimization problems, solved by relaxation methods as well. The first one consists of the estimation of the separating matrix when the autoregressive structure of the sources is fixed. The second one aims at estimating this structure when the separating matrix is fixed. We show that the first problem is equivalent to the determinant maximization of the separating matrix under nonlinear constraints. We prove the existence and the consistency of the maximum likelihood estimator. We also give the expression of Fisher's information matrix. Then, we study, by computer simulations, the performance of our estimator and show the improvement of its achievements w.r.t. both quasimaximum likelihood and second-order blind identification (SOBI) estimators.