A new domain decomposition method with Lagrange multipliers for elliptic problems is introduced. It is based on a reformulation of the well--known FETI method as a saddle point problem with both primal and dual variables as unknowns. The resulting linear system is solved with block--structured preconditioners combined with a suitable Krylov subspace method. This approach allows the use of inexact subdomain solvers for the positive definite subproblems. It is shown that the condition number of the preconditioned saddle point problem is bounded independently of the number of subregions and depends only polylogarithmically on the number of degrees of freedom of individual local subproblems. Numerical results are presented for a plane stress cantilever membrane problem.