  Families of stable curves of genus $\gamma$ over a smooth curve $C$
correspond to morphisms from $C$ to the moduli stack of stable curves
$\bar{\cal M}_\gamma$. It is natural to compactify the corresponding moduli
problem using stable maps into the stack. In order to get a complete moduli
problem, the source curves must acquire extra structure, and you will have to
read the paper to find what that is.
  In this paper, we define these stable maps into $\bar{\cal M}_\gamma$ in
characteristic 0, and show that they form a proper Deligne-Mumford stack
admitting a projective coarse moduli space. A comparison with Alexeev's work is
given.
  Natural generalizations for stable maps into other Deligne-Mumford stacks
will appear in a follow up paper, along with applications to areas such as
admissible covers, level structures and higher dimensional families.
