  We show that the one- and two-loop $\beta$-functions of the closed, bosonic
string can be written in a manifestly O($D$,$D$)-covariant form. Based on this
result, we prove that 1) Poisson-Lie symmetric $\sigma$-models are two-loop
renormalisable and 2) their $\beta$-functions are invariant under Poisson-Lie
T-duality. Moreover, we identify a distinguished scheme in which Poisson-Lie
symmetry is manifest. It simplifies the calculation of two-loop
$\beta$-functions significantly and thereby provides a powerful new tool to
advance into the quantum regime of integrable $\sigma$-models and generalised
T-dualities. As an illustrating example, we present the two-loop
$\beta$-functions of the integrable $\lambda$- and $\eta$-deformation.
