The quantum dynamics of a quartic double well, subjected to a harmonically oscillating field, is studied in the framework of the Floquet formalism. The modifications of the familiar tunneling process due to the driving are investigated numerically and explained in terms of the structure of the corresponding local quasienergy spectrum. In particular, there is a one-dimensional manifold in the parameter space spanned by the amplitude and frequency of the driving force, where tunneling is almost completely suppressed by the coherent driving. The quantal dynamics in the semiclassical regime as well as the influence of weak incoherent processes are briefly discussed.