This　paper　is　concerned　with　efficient　algorithms　for　solving　Weber　problem,　which　is　an　important　problem　arising　in　the　facility　location　problems.　In　this　paper,　we　reformulate　the　Weber　problem　as　its　equivalent　form　and　then　propose　a　semismooth　Newton　based　augmented　Lagrangian　(SSNAL)　algorithm　for　solving　Weber　problem.　The　global　convergence　and　locally　asymptotically　superlinear　convergence　of　the　SSNAL　algorithm　are　characterized　under　mild　conditions.　Numerical　experiments　conducted　on　synthetic　data　sets　demonstrate　that　the　SSNAL　algorithm　outperforms　several　state-of-the-art　algorithms　in　terms　of　efficiency　and　robustness.