  We compute the AdS Veneziano amplitude for type IIB gluon scattering in
$AdS_5 \times S^3$ to all orders in $\alpha'$ in a small curvature expansion.
This is achieved by combining a dispersion relation in the dual $4d$
$\mathcal{N}=2$ SCFT with an ansatz for the amplitude as a worldsheet integral
in terms of multiple polylogarithms. The first curvature correction is fully
fixed in this way and satisfies consistency checks in the high energy limit,
the low energy expansion as previously fixed using supersymmetric localisation,
and for the energy of massive string operators, which we independently compute
using a semiclassical expansion. We also combine localisation with this first
curvature correction to fix the unprotected $D^4F^4$ correction to the
amplitude at finite curvature.
