How's that? Too thick? I'm sure there's a better way, or you can play with my parameters:
1. Wand select the white at 50 tolerance and inverse the selection. This is so you've got the line on a transparent layer without the white.
2. Contextual menu > new layer via copy. Duplicate your new layer. (You should have two copies.)
3. Select Menu > Modify > Expand. Expand it by 1 pixel.
4. Gaussian Blur both of your new layers by 1 pixel, then merge them.
5. Unsharp Mask the new layer with an amount of 80, radius of 40, and threshhold of 10.
Suppose you could fiddle with that process to make it less thick, but at least it's thicker like you wanted and anti-aliased to an extent.
