  In local quantum field theory, the entanglement entropy of a region is
divergent due to the arbitrary short-wavelength correlations near the boundary
of the region. Quantum gravitational fluctuations are expected to cut off the
entropy of the ultraviolet modes. Guided by this, we study the entanglement
entropy in closed string theory. We factorize the configuration space of closed
strings based on the position of the center of mass. We then compute the
one-loop R\'enyi partition functions corresponding to this factorization using
the replica method and string field theory. The short-wavelength modes are cut
off at the string scale and the one-loop entanglement entropy is
ultraviolet-finite. A non-trivial path integration measure, required to produce
the correct one-loop vacuum amplitude, plays a key role.
