  We use Morse theory to prove that the Lefschetz Hyperplane Theorem holds for
compact smooth Deligne-Mumford stacks over the site of complex manifolds. For
$Z \subset X$ a hyperplane section, $X$ can be obtained from $Z$ by a sequence
of deformation retracts and attachments of high-dimensional finite disc
quotients. We use this to derive more familiar statements about the relative
homotopy, homology, and cohomology groups of the pair $(X,Z)$. We also prove
some preliminary results suggesting that the Lefschetz Hyperplane Theorem holds
for Artin stacks as well. One technical innovation is to reintroduce an
inequality of {\L}ojasiewicz which allows us to prove the theorem without any
genericity or nondegeneracy hypotheses on $Z$.
