  We consider so-called Yang-Baxter deformations of bosonic string
sigma-models, based on an $R$-matrix solving the (modified) classical
Yang-Baxter equation. It is known that a unimodularity condition on $R$ is
sufficient for Weyl invariance at least to two loops (first order in
$\alpha'$). Here we ask what the necessary condition is. We find that in cases
where the matrix $(G+B)_{mn}$, constructed from the metric and $B$-field of the
undeformed background, is degenerate the unimodularity condition arising at one
loop can be replaced by weaker conditions. We further show that for
non-unimodular deformations satisfying the one-loop conditions the Weyl
invariance extends at least to two loops (first order in $\alpha'$). The
calculations are simplified by working in an $O(D,D)$-covariant doubled
formulation.
