  We study the linearized stability of n-vortex solutions of the magnetic
Ginzburg-Landau (or Abelian-Higgs) equations. We prove that the fundamental
vortices (n=1,-1) are stable for all values of the coupling constant, k, and we
prove that the higher-degree vortices (|n| > 1) are stable for k < 1 and
unstable for k > 1. This resolves a long-standing conjecture.
