  We show that a $C^\star$-algebra $B$ contains a dense left or right Fr\'echet
ideal $A$, with $A$ a nuclear locally convex space, if and only if the
primitive ideal space Prim$(B)$ of $B$ is discrete and countable, and $B/I$ is
finite dimensional for each $I \in $ Prim$(B)$. We show the forward implication
holds for a general Banach algebra $B$, if the ideal is assumed two-sided. For
$C^\star$-algebras, we construct all two-sided dense nuclear ideals by defining
a set of matrix-valued Schwartz functions on the countable discrete space
Prim$(B)$.
