The intersection local time is the Hausdorff measure on the intersection of two independent Brownian paths in dimensions two or three. We show that, although the multifractal formalism predicts a trivial spectrum, the multifractal spectrum is actually nontrivial at the thin end. The spectrum is a convex function, which is given in terms of the Brownian intersection exponents recently studied by Lawler, Schramm and Werner. This is joint work with Achim Klenke (Mainz).