We consider the incompressible Navier-Stokes equations in a two-dimensional exterior domain Ω, with no-slip boundary conditions. Our initial data are of the form u 0 = αΘ 0 + v 0 , where Θ 0 is the Oseen vortex with unit circulation at infinity and v 0 is a solenoidal perturbation belonging to L 2 (Ω) 2 ∩ L q (Ω) 2 for some q ∈ (1, 2). If α ∈ R is sufficiently small, we show that the solution behaves asymptotically in time like the self-similar Oseen vortex with circulation α. This is a global stability result, in the sense that the perturbation v 0 can be arbitrarily large, and our smallness assumption on the circulation α is independent of the domain Ω.