  We consider the low-energy effective action of string theory at order $\alpha
'$, including $R^2$-corrections to the Einstein-Hilbert gravitational action
and non-trivial dilaton coupling. By means of a convenient field redefinition,
we manage to express the theory in a frame that enables us to solve its field
equations analytically and perturbatively in $\alpha ' $ for a static
spherically symmetric ansatz in an arbitrary number of dimensions. The set of
solutions we obtain is compatible with asymptotically flat geometries
exhibiting a regular event horizon at which the dilaton is well-behaved. For
the 4-dimensional case, we also derive the stationary black hole configuration
at first order in $\alpha '$ and in the slowly rotating approximation. This
yields string theory modifications to the Kerr geometry, including terms of the
form $a$, $a^2$, $\alpha '$ and $a\alpha '$. In addition, we obtain the first
$\alpha'$ correction to the C-metrics, which accommodates accelerating black
holes. We work in the string frame and discuss the connection to the Einstein
frame, for which rotating black holes have already been obtained in the
literature.
