  We investigate the universal Jacobian of degree n line bundles over the
Hurwitz stack of double covers of P^1 by a curve of genus g. Our main results
are: the construction of a smooth, irreducible, universally closed (but not
separated) moduli compactification of this universal Jacobian; a description of
the global geometry and moduli properties of these stacks; and a computation of
the Picard groups of these stacks in the cases when n-g is even. An important
ingredient of our work is the parametrization of line bundles on double covers
by binary quadratic forms. This parametrization generalizes the classical
number theoretic correspondence between ideal classes of quadratic rings and
integral binary quadratic forms, which in particular gives the group law on
integral binary quadratic forms first discovered by Gauss.
