Just to close the book on this one...
John, the "zero-g" comment was a trick answer. Yes, at the centre of the Sun, it's mass is equally distributed all around you so the net gravitational field contribution from the Sun is zero. But time dilation due to gravity isn't proportional to the gravitational field but the gravitational potential field (a subtle difference that trips up a lot of people).
The GPF (P) = GM/d, i.e. it is proportional to the mass, but falls off linearly with distance, not with the square of the distance like gravity (g). This has a very profound effect. Because P falls off a lot more slowly that g, even something massive and nearby like the Earth has only a minor contribution to the overall value of P compared to the rest of the Milky Way, so the decrease in time dilation between the surface of the Earth and orbit is very small.
Big point (and small confession), P (the gravitational potential field) is a scalar, not a vector, so being at the centre of a massive body does not mean that having an equal proportion of that mass pulling in every direction "cancels out" like it does for gravity. So P continues to increase below the surface and reaches a maximum at the centre, but unless you know how the mass is distributed around the body (i.e. how it's density varies), you can't do an exact calculation.
As I don't know the density of all the different layers of the Sun (does anyone?), I don't in fact know the correct answer like I said, I just know how to get it.
Lecture's over! Let's get back to roasting the fundtards!
(P.S. if anyone spots a mistake in the above, do point it out. I'm rusty as hell at this stuff and have probably made several!)