Complicated cosmic string loops will fragment until they reach simple, non-intersecting ("stable") configurations. Through extensive numerical study, these attractor loop shapes are characterized including their length, velocity, kink, and cusp distributions. An initial loop containing $M$ harmonic modes will, on average, split into 3M stable loops. These stable loops are approximately described by the degenerate kinky loop, which is planar and rectangular, independently of the number of modes on the initial loop. This is confirmed by an analytic construction of a stable family of perturbed degenerate kinky loops. The average stable loop is also found to have a 40% chance of containing a cusp. This new analytic scheme explicitly solves the string constraint equations.