The inextensibility constraint is encountered in many physical problems involving thin solids interacting with a fluid. It is generally imposed in numerical simulations by means of a penalty method. Here, we propose a novel saddle-point approach allowing to impose it through a Lagrange multiplier defined on the thin structure, the tension. The functional analysis of the problem allows to determine which boundary conditions are needed for this problem. The forces originating from the structure appear as a boundary condition for the fluid problem, defined on a moving boundary which represents the structure. The problem is discretised with mixed finite elements. The mesh of the thin solid is included in the mesh of the bulk, and is advected by its velocity in the course of time iterations. The appropriate choice of the finite element spaces for this mixed approach is discussed and it is shown that boundary conditions on the thin structure edges impact on this choice. Numerical tests are performed which demonstrate the convergence and robustness of the method. These examples include the simulation of a closed, membrane bound object (a vesicle) and of a filament or flag in a large Reynolds-number flow.