Time dilation layers

A test mass at the center of the earth is weightless. Does a clock at the center of the earth experience no gravitational time dilation due to general relativity? If so, what layer of the earth, at what depth, experiences the greatest time dilation? What if the planet's density is not uniform (but still spherically symmetric) like Jupiter?

A Newtonian approximation will probably do fine. Given the radial density profile, calculate the strength of gravity at each point. For uniform density, mass increases by the cube, but gravity decreases by the inverse square, so the force increases linearly, so the maximum time dilation is at the surface.

Update: http://en.wikipedia.org/wiki/File:Earth-G-force.png