In this article a primal barrier interior-point method for moving horizon estimation (MHE) is presented. It exploits the structure of the KKT systems yielding an algorithm with linear complexity in the horizon length as opposed to cubically as in unstructured solvers. Ideas of square root covariance Kalman filtering are proposed in order to update covariance matrices occurring in the factorization of the KKT matrix efficiently and in a numerically stable way. The algorithm is able to compute - without any additional costs - the covariance of the last estimate within the horizon, which reflects the accuracy of the estimate.