We consider the so-called Babuška method of finite elements with Lagrange multipliers for numerically solving the problem Δ u = f \Delta u = f in Ω \Omega , u = g u = g on ∂ Ω \partial \Omega , Ω ⊂ R n \Omega \subset {R^n} , n ⩾ 2 n \geqslant 2 . We state a number of local conditions from which we prove the uniform stability of the Lagrange multiplier method in terms of a weighted, mesh-dependent norm. The stability conditions given weaken the conditions known so far and allow mesh refinements on the boundary. As an application, we introduce a class of finite element schemes, for which the stability conditions are satisfied, and we show that the convergence rate of these schemes is of optimal order.