  The main result of this article is that the component of the
Alexeev-Koll\'{a}r-Shepherd-Barron moduli space of stable surfaces
parameterizing stable degenerations of symmetric squares of curves is
isomorphic to the moduli space of stable curves. The moduli space of stable
curves is used to ensure that any degeneration can be replaced by a
degeneration to the symmetric square of a stable curve. These degenerations are
not stable, however, but are partially resolved by the relative Hilbert scheme
of length two subschemes in the fibers of a family of stable curves. The
article gives a geometric description of the relative minimal models of these
Hilbert schemes.
