  Let X be a smooth Mori dream space of dimension at least 4. We show that, if
X satisfies a suitable GIT condition which we call "small unstable locus", then
every smooth ample divisor Y of X is also a Mori dream space. Moreover, the
restriction map identifies the Neron-Severi spaces of X and Y, and under this
identification every Mori chamber of Y is a union of some Mori chambers of X,
and the nef cone of Y is the same as the nef cone of X. This Lefschetz-type
theorem enables one to construct many examples of Mori dream spaces by taking
"Mori dream hypersurfaces" of an ambient Mori dream space, provided that it
satisfies the GIT condition. To facilitate this, we then show that the GIT
condition is stable under taking products and taking the projective bundle of
the direct sum of at least three line bundles, and in the case when X is toric,
we show that the condition is equivalent to the fan of X being 2-neighborly.
