The weighted nonlinear least-squares problem for low-rank signal estimation is considered. The problem of constructing a numerical solution that is stable and fast for long time series is addressed. A modified weighted Gauss–Newton method, which can be implemented through the direct variable projection onto a space of low-rank signals, is proposed. For a weight matrix that provides the maximum likelihood estimator of the signal in the presence of autoregressive noise of order p the computational cost of iterations is (Formula presented.) as N tends to infinity, where N is the time-series length, r is the rank of the approximating time series. Moreover, the proposed method can be applied to data with missing values, without increasing the computational cost. The method is compared with state-of-the-art methods based on the variable projection approach in terms of floating-point numerical stability and computational cost.