  We consider the well-known solution of the Heterotic Superstring effective
action to zeroth order in $\alpha'$ that describes the intersection of a
fundamental string with momentum and a solitonic 5-brane and which gives a
3-charge, static, extremal, supersymmetric black hole in 5 dimensions upon
dimensional reduction on $\mathrm{T}^{5}$. We compute explicitly the
first-order in $\alpha'$ corrections to this solution, including
$\mathrm{SU}(2)$ Yang-Mills fields which can be used to cancel some of these
corrections and we study the main properties of this $\alpha'$-corrected
solution: supersymmetry, values of the near-horizon and asymptotic charges,
behavior under $\alpha'$-corrected T-duality, value of the entropy (using Wald
formula directly in 10 dimensions), existence of small black holes etc. The
value obtained for the entropy agrees, within the limits of approximation, with
that obtained by microscopic methods. The $\alpha'$ corrections coming from
Wald's formula prove crucial for this result.
