  We confirm the leading $\alpha'^3$ correction to the 4d, $\mathcal N = 1$
K\"{a}hler potential of type IIB orientifold compactifications, proportional to
the Euler characteristic of the Calabi-Yau threefold (BBHL correction). We
present the explicit solution for the $\alpha'^3$-modified internal background
metric in terms of the non-harmonic part of the third Chern form of the leading
order Calabi-Yau manifold. The corrected internal manifold is almost Calabi-Yau
and admits an $SU(3)$ structure with non-vanishing torsion. We also find that
the full ten-dimensional Einstein frame background metric is multiplied by a
non-trivial Weyl factor. Performing a Kaluza-Klein reduction on the modified
background we derive the $\alpha'^3$-corrected kinetic terms for the dilaton
and the K\"{a}hler deformations of the internal Calabi-Yau threefold for
arbitrary $h^{1,1}$. We analyze these kinetic terms in the 4d, $\mathcal N = 2$
un-orientifolded theory, confirming the expected correction to the K\"ahler
moduli space prepotential, as well as in the 4d, $\mathcal N = 1$ orientifolded
theory, thus determining the corrections to the K\"ahler potential and K\"ahler
coordinates.
