  We characterize the Hilbert--Schmidt class membership of commutator with the
Hilbert transform in the two weight setting. The characterization depends upon
the symbol of the commutator being in a new weighted Besov space. This follows
from a Schatten class $S_p$ result for dyadic paraproducts, where $1< p <
\infty $. We discuss the difficulties in extending the dyadic result to the
full range of Schatten classes for the Hilbert transform.
