  We express one-loop string amplitudes involving both open and closed strings
as sum over pure open string amplitudes. These findings generalize the
analogous tree-level result to higher loops and extend the tree-level
observation that in gravitational amplitudes a graviton can be traded for two
gluons. Our results are derived from analytic continuation of closed string
world-sheet coordinates on the cylinder resulting in pairs of real open string
coordinates located at the two cylinder boundaries subject to a one-loop
kernel. The latter depends on the loop momentum flowing between the two
cylinder boundaries and relates to intersection theory for twisted cycles.
Finally, contact is made with one-loop open string monodromy relations. The
latter contain a boundary term, which is related to non-physical contours on
the cylinder. A physical interpretation of the latter in terms of a closed
string insertion is given.
