We consider the following problem: estimate the Lipschitz continuous diffusion coefficient σ 2 from the path of a 1-dimensional diffusion process sampled at times i/n,i=0,...,n, when we believe that σ 2 actually belongs to a smaller regular parametric set Σ 0 . By introducing random normalizing factors in the risk function, we obtain confidence sets which can be essentially better than the minimax rate n - 1 / 3 of estimation for Lipschitz functions in diffusion models. With a prescribed confidence level α n , we show that the best possible attainable (random) rate is (logα n - 1 /n) 2 / 5 . We construct an optimal estimator and an optimal random normalizing factor in the sense of Lepski (1999).This has some consequences for classical estimation: our procedure is adaptive w.r.t. Σ 0 and enables us to test the hypothesis that σ 2 is parametric against a family of local alternatives with prescribed 1st and 2nd-type error probabilities.