  We present a second-order-in-time finite difference scheme for the
Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable
and unconditionally energy stable. At each time step, this scheme leads to a
system of nonlinear equations that can be efficiently solved by a nonlinear
multigrid solver. Owing to the energy stability, we derive an $\ell^2 (0,T;
H_h^3)$ stability of the numerical scheme. To overcome the difficulty
associated with the convection term $\nabla \cdot (\phi \boldsymbol{u})$, we
perform an $\ell^\infty (0,T; H_h^1)$ error estimate instead of the classical
$\ell^\infty (0,T; \ell^2)$ one to obtain the optimal rate convergence
analysis. In addition, various numerical simulations are carried out, which
demonstrate the accuracy and efficiency of the proposed numerical scheme.
