  We study the symmetry of the one-loop effective action of bosonic string
theory under non-Abelian T-duality transformations. It is shown that the
original Lagrangian and its dual are proportional. This result implies that the
corresponding reduced low energy effective actions are equivalent and leads to
a functional relation between the Weyl anomaly coefficients of the original and
dual two-dimensional non-linear sigma models. {}Finally, we apply this
formalism to some simple examples.
