Passive scalar turbulence forced steadily is characterized by the velocity correlation scale L, injection scale l, and diffusive scale rd. The scales are well separated if the diffusivity is small, rd≪l,L, and one normally says that effects of diffusion are confined to smaller scales, r≪rd. However, if the velocity is single scale, one finds that a weak dependence of the scalar correlations on the molecular diffusivity persists to even larger scales, e.g., l≫r≫rd (E. Balkovsky , Pis'ma Zh. Eksp. Teor. Fiz. 61, 1012 (1995) [JETP Lett. 61, 1049 (1995)]). We consider the case of L≫l and report a counterintuitive result, namely the emergence of a new range of large scales, L≫;r≫l2/ rd, where the diffusivity shows a strong effect on scalar correlations.