We show that, despite conventional belief, a weak perturbation in a Hamiltonian system may lead to a wide chaotic layer and fast chaotic transport. This occurs in a spacially periodic Hamiltonian system subject to a dipole‐type time‐periodic perturbation with a small frequency. We explain this and develop an explicit theory for the layer width, in nice agreement with simulations.The challenging unsolved problems relate to the chaotic transport in the system and to numerous applications, in particular to the noise‐induced diffusion of particles on crystalline surfaces.