This is actually an interesting question for the nomenclature people- is it a planet or a star? The thing is essentially solid, but so are neutron "stars", and while it's the size of a planet it used the be the size of a star.
well, that comes down to whether theres fusion going on inside
I would say it's not a star anymore, because without it's outer layers it shouldn't be hot enough to fuse carbon. I dont think the 'definition' of 'planet' took into account this situation, but I dont beleive theres anything in the wording preventing it. Planets were once particles of dust, afterall, so what we call it now does not take into account what we might have called it in the past.
Diamond planets have been suggested before, and are probably the fate of white dwarfs after they cool to black dwarfs; although no one knows as it would take longer than the universe has been around to see that happen. FYI, the fate of our sun is to become a white dwarf. So... if your girl can wait.... for a really, really long time.... you can get her a really big diamond
On another note, I think getting a visible-light picture of a neutron star with details should be of prime importance. It's the closest thing we can find to a black hole and also is dependant on quantum-mechanical principles. I believe they generally emit in the radio spectrum, so I would guess that its possible to see one without tremendous glare.