A Leibniz algebra is a vector space equipped with a bracket satisfying [x,[y,z]] = [[x,y],z] - [[x,z],y]. A Lie algebra is an example of a Leibniz algebra. A dual Leibniz algebra is a vector space equipped with a product satisfying ((ab)c) = (a(bc)) + (a(cb)). Dual is taken in the operad sense, so the tensor product of a Leibniz algebra by a dual Leibniz algebra is a Lie algebra. In this paper it is proved that the Leibniz cohomology HL^*(g) of the Leibniz algebra g is equipped with a cup-product which makes it into a graded dual Leibniz algebra.