The paper is devoted to the solution of a weighted non-linear least-squares problem for low-rank signal estimation, which is related Hankel structured low-rank approximation problems. The solution is constructed by a modified weighted Gauss-Newton method. The advantage of the suggested method is the possibility of its stable and fast implementation. The method is compared with a known method, which uses the variable-projection approach, by stability, accuracy and computational cost. For the weighting matrix, which corresponds to autoregressive processes of order $p$, the computational cost is $O(N r^2 + N p^2 + r N \log N)$, where $N$ is the time series length, $r$ is the rank of approximating time series. For the proof of the suggested method, useful properties of the space of series of rank $r$ are studied.