  Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey
number} to be the minimum number of vertices $n$ such that any $k$-edge
coloring of the complete graph $K_n$ contains either a rainbow (all different
colored) triangle or a monochromatic copy of $G$. In this paper, we obtain the
exact value of the Gallai-Ramsey numbers for the union of two stars in many
cases and bounds in other cases. This work represents the first class of
disconnected graphs to be considered as the desired monochromatic subgraph.
