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In this paper we devise and analyze a mixed finite element method for a modified Cahn--Hilliard equation coupled with a nonsteady Darcy--Stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable. We prove that the discrete phase variable is bounded in $L logical or infty \left(0,T; L logical or infty\right)$ and the discrete chemical potential is bounded in $L logical or infty \left(0,T; L arrow up \right)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions.