A hybrid algorithm consisting of a Gauss--Newton method and a second-order method for solving constrained and weighted nonlinear least squares problems is developed, analyzed, and tested. One of the advantages of the algorithm is that arbitrarily large weights can be handled and that the weights in the merit function do not get unnecessarily large when the iterates diverge from a saddle point. The local convergence properties for the Gauss--Newton method are thoroughly analyzed and simple ways of estimating and calculating the local convergence rate for the Gauss--Newton method are given. Under the assumption that the constrained and weighted linear least squares subproblems attained in the Gauss--Newton method are not too ill conditioned, global convergence towards a first-order KKT point is proved.