  It has recently been shown that a set of the generalized type IIB
supergravity equations follows from the requirement of kappa symmetry of the
type IIB Green-Schwarz superstring theory defined on an arbitrary background.
In this paper, we show that the whole bosonic part of the generalized type II
supergravity equations can be reproduced from the T-duality covariant equations
of motion of the double field theory by choosing a non-standard solution of the
strong constraint. Then, by using the doubled formalism, we show the Weyl
invariance of the bosonic string sigma model on a generalized gravity
background. According to the dual-coordinate dependence of the dilaton, the
Fradkin-Tseytlin term nicely removes the Weyl anomaly. This result seems likely
to support that string theories can be consistently defined on arbitrary
generalized supergravity backgrounds.
