We consider a fully practical finite element approximation of the degenerate Cahn-Hilliard equation with elasticity: Find the conserved order parameter, , and the displacement field, , such that {}{ t}= (b() [- + ^{-1}'() + 12 c' (){ C}{{ E}}( u) : {{ E}}( u)] ), (c(){ C} {{ E}}({u})) = 0, subject to an initial condition on and boundary conditions on both equations. Here is the interfacial parameter, is a nonsmooth double well potential, is the symmetric strain tensor, is the possibly anisotropic elasticity tensor, with and is the degenerate diffusion mobility. In addition to showing stability bounds for our approximation, we prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system in two space dimensions. Finally, some numerical experiments are presented.