  A solution of the linearized Einstein and Nambu-Goto equations is constructed
which describes the evaporation of a certain type of rotating cosmic string -
the Allen-Casper-Ottewill loop - under the action of its own self-gravity. The
solution evaporates self-similarly, and radiates away all its mass-energy and
momentum in a finite time. Furthermore, the corresponding weak-field metric can
be matched to a remnant Minkowski spacetime at all points on the future light
cone of the final evaporation point of the loop.
