  We evaluate one-loop open-string amplitudes at finite $\alpha'$ for the first
time. Our method involves a deformation of the integration contour over the
modular parameter $\tau$ to a fractal contour introduced by Rademacher in the
context of analytic number theory. This procedure leads to explicit and
practical formulas for the one-loop four-point amplitudes in type-I superstring
theory, amenable to numerical evaluation. We plot the amplitudes as a function
of the Mandelstam invariants $s$ and $t$ and directly verify long-standing
conjectures about their behaviour at high energies.
