  We show for the first time that a {\it weak} perturbation in a Hamiltonian
system may lead to an arbitrarily {\it wide} chaotic layer and {\it fast}
chaotic transport. This {\it generic} effect occurs in any spatially periodic
Hamiltonian system subject to a sufficiently slow ac force. We explain it and
develop an explicit theory for the layer width, verified in simulations.
Chaotic spatial transport as well as applications to the diffusion of particles
on surfaces, threshold devices and others are discussed.
