Stationary and axially symmetric metrics play an important role in astrophysics. Weyl metrics form the static vacuum sub-class of such metrics, which contains the Schwarzschild metric as its most prominent example. Killing tensor fields play a crucial role in physics since they correspond to invariants of the geodesic flow. We study the space of Killing tensor fields of valence 3 for Weyl metrics and prove that any involutive cubic Killing tensor field is generated by Killing vector fields and quadratic Killing tensor fields.