  In this paper we show that the hydrodynamic problem for three-dimensional
water waves with strong surface-tension effects admits a fully localised
solitary wave which decays to the undisturbed state of the water in every
horizontal direction. The proof is based upon the classical variational
principle that a solitary wave of this type is a critical point of the energy
subject to the constraint that the momentum is fixed. We prove the existence of
a minimiser of the energy subject to the constraint that the momentum is fixed
and small. The existence of a small-amplitude solitary wave is thus assured,
and since the energy and momentum are both conserved quantities a standard
argument may be used to establish the stability of the set of minimisers as a
whole. `Stability' is however understood in a qualified sense due to the lack
of a global well-posedness theory for three-dimensional water waves.
