  Llarull's theorem characterizes the round sphere $S^n$ among all spin
manifolds whose scalar curvature is bounded from below by $n(n-1)$. In this
paper we show that if the scalar curvature is bounded from below by
$n(n-1)-\varepsilon$, the underlying manifold is $C^0$-close to a finite number
of spheres outside a small bad set. This completely solves Gromov's spherical
stability problem.
