We consider a fully practical finite element approximation of the degenerate Cahn-Hilliard equation with elasticity: Find the conserved order parameter, θ(x,t)∈[−1,1], and the displacement field, u​(x,t)∈R2, such that γ∂t∂θ​=∇⋅(b(θ)∇[−γΔθ+γ−1Ψ′(θ)+21​c′(θ)CE​​(u​):E​​(u​)]),∇⋅(c(θ)CE​​(u​))=0​, subject to an initial condition θ0(⋅)∈[−1,1] on θ and boundary conditions on both equations. Here γ∈R>0​ is the interfacial parameter, Ψ is a nonsmooth double well potential, E​​ is the symmetric strain tensor, C is the possibly anisotropic elasticity tensor, c(s):=c0​+21​(1−c0​)(1+s) with c0​(γ)∈R>0​ and b(s):=1s2 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