The system of equations resulting from a mixed finite element approximation of the first biharmonic boundary value problem is solved by various preconditioned Uzawa-type iterative methods. The preconditioning matrices are based on simple finite element approximations of the Laplace operator and some factorizations of the corresponding matrices. The most efficient variants of these iterative methods require asymptoticallyO(h−0,5In e−1) iterations andO(h−p−0,5In e−1) arithmetic operations only, where e denotes the relative accuracy andh is a mesh-size parameter such that the number of unknowns grows asO(h−p),h→0.