  We develop the perturbation theory of double field theory around arbitrary
solutions of its field equations. The exact gauge transformations are written
in a manifestly background covariant way and contain at most quadratic terms in
the field fluctuations. We expand the generalized curvature scalar to cubic
order in fluctuations and thereby determine the cubic action in a manifestly
background covariant form. As a first application we specialize this theory to
group manifold backgrounds, such as $SU(2) \simeq S^3$ with $H$-flux. In the
full string theory this corresponds to a WZW background CFT. Starting from
closed string field theory, the cubic action around such backgrounds has been
computed before by Blumenhagen, Hassler and L\"ust. We establish precise
agreement with the cubic action derived from double field theory. This result
confirms that double field theory is applicable to arbitrary curved background
solutions, disproving assertions in the literature to the contrary.
