  We develop a method to compute $H^2$-conforming finite element approximations
in both two and three space dimensions using readily available finite element
spaces. This is accomplished by deriving a novel, equivalent mixed variational
formulation involving spaces with at most $H^1$-smoothness, so that conforming
discretizations require at most $C^0$-continuity. The method is demonstrated on
arbitrary order $C^1$-splines.
