After reviewing the double copy structure of tree-level and one-loop gravitational amplitudes we discuss relations between closed and open string amplitudes at one-loop. While at tree-level these relations are known as Kawai-Lewellen-Tye (KLT) and/or double copy relations, here we investigate how such relations are manifested at one-loop. Although, we find examples of one-loop closed string amplitudes, that can strikingly be written as sum over squares of one-loop open string amplitudes, generically the one-loop closed string amplitudes assume a form reminiscent from the (conjectural) one-loop doubly copy structure of gravitational amplitudes involving a loop momentum. This double copy structure represents the one-loop generalization of the KLT relations. The latter give rise to a closed/open string correspondence linking closed and open string amplitudes at one-loop. Likewise, in the field-theory limit we obtain of proof of the field-theory double copy conjecture.