Melting diamond and cool down as diamond

While I agree in principle with David Lynch's answer, I think it's good to take a closer look at the phase diagram (adapted from http://upload.wikimedia.org/wikipedia/commons/4/46/Carbon_basic_phase_diagram.png):

I added the arrows to show possible paths you might follow.

Red path: diamond would become graphite before melting; the molten carbon becomes diamond just above 10 GPa, and you cool it down while maintaining the pressure. Once the diamond is cool enough it can be depressurised slowly without changing phase (the hashed region has to be traversed carefully).

Blue path: if you just heat your diamond, it will turn to graphite and then vaporize (sublimate) around 4000 K.

The green path shows the only "sure" way to melt diamond - starting at a very high pressure, then raising the temperature; above 5000 K one could either continue raising the temperature, or lower the pressure.

Note that there is a real problem with doing this - there are no containers that I know of that will allow that combination of temperature and pressure to be maintained. Synthetic diamonds have been made, but typically not by growing from the melt...

Carbon, at atmospheric pressure, has no melting point as its triple point is at 10.8 ± 0.2 MPa and 4,600 ± 300 K (~4,330°C or 7,820°F), so it sublimes at about 3,900 K. You can make it liquid at larger pressures, in an inert gas atmosphere. When you cool it, what it becomes depends on the pressure. Below about 10 GPa it will become graphite, above that it will become diamond again. You can see a carbon phase diagram here.