  We prove that the disordered monomer-dimer model does not admit infinite
volume incongruent ground states in $\mathbb{Z}^d$ which can be obtained as a
limit of finite volume ground states. Furthermore, we also prove that these
ground states are stable under perturbation of the weights in a precise sense.
  As an application, we obtain a CLT for the ground state weight for a growing
sequence of tori. Our motivation stems from a similar and long standing open
question for the short range Edwards-Anderson spin glass model.
