A reduced, cancellative, torsion-free, commutative monoid M can be embedded in an integral domain R, where the atoms (irreducible elements) of M correspond to a subset of the atoms of R. This fact was used by J. Coykendall and B. Mammenga to show that for any reduced, cancellative, torsion-free, commutative, atomic monoid M, there exists an integral domain R with atomic factorization structure isomorphic to M. More generally, we show that any “nice” subset of atoms of R can be realized as the set of atoms of an integral domain T that contains R. We will also give several applications of this result.