  In this Letter, we provide evidence for a new double-copy structure in
one-loop amplitudes of the open superstring. Their integrands with respect to
the moduli space of genus-one surfaces are cast into a form where
gauge-invariant kinematic factors and certain functions of the punctures --
so-called generalized elliptic integrands -- enter on completely symmetric
footing. In particular, replacing the generalized elliptic integrands by a
second copy of kinematic factors maps one-loop open-string correlators to
gravitational matrix elements of the higher-curvature operator R^4.
