We consider the infinite-dimensional Lie group G which is the semidirect product of the group of compactly supported diffeomorphisms of a Riemannian manifold X and the commutative multiplicative group of functions on X. The group G naturally acts on the space M(X) of Radon measures on X. We would like to define a Laplace operator associated with a natural representation of G in L2(M(X),μ). Here μ is assumed to be the law of a measure-valued Lévy process. A unitary representation of the group cannot be determined, since the measure μ is not quasi-invariant with respect to the action of the group G. Consequently, operators of a representation of the Lie algebra and its universal enveloping algebra (in particular, a Laplace operator) are not defined. Nevertheless, we determine the Laplace operator by using a special property of the action of the group G (a partial quasi-invariance). We further prove the essential self-adjointness of the Laplace operator. Finally, we explicitly construct a diffusion process on M(X) whose generator is the Laplace operator.