  We construct a lattice action for ${\cal N}=4$ super Yang-Mills theory in
four dimensions which is local, gauge invariant, free of spectrum doubling and
possesses a single exact supersymmetry. Our construction starts from the
observation that the fermions of the continuum theory can be mapped into the
component fields of a single real anticommuting Kahler-Dirac field. The
original supersymmetry algebra then implies the existence of a nilpotent scalar
supercharge $Q$ and a corresponding set of bosonic superpartners. Using this
field content we write down a $Q$-exact action and show that, with an
appropriate change of variables, it reduces to a well-known twist of ${\cal
N}=4$ super Yang-Mills theory due to Marcus. Using the discretization
prescription developed in an earlier paper on the ${\cal N}=2$ theory in two
dimensions we are able to translate this geometrical action to the lattice.
