cs6 - Is it possible to make a gradient follow a path in Illustrator?

If you take this image:

Is it possible to make a gradient in Illustrator that follows that path? I made something similar (just not with a single path, but with the pen tool) and I want to have the gradient follow the shape, but instead it just goes either linear (top to bottom) or radial (from the inside out). I want to have the start of the spiral lighter and than the tip in the center to be darker.