Is it possible to melt a diamond?
The melting point of diamond is certainly an exotic topic. Diamond is slightly less stabile than its allotrope, graphite, and will in most circumstances turn to graphite way before its melting point. If you have enough pressure it will stay diamondlike and it will at some point melt.

A diamond is not a single molecule unless you stretch the definition of molecule. It is usually defined as the smallest fundamental unit, and for diamond I'd call the tetragonal carbon unit cell the smallest fundamental unit, or maybe even a carbon atom. Diamond, having one of the highest values for heat transfer among substances would certainly melt almost like a single molecule though. But in theory, one and one carbon decides to leave behind it's friends bound by the diamond structure boundaries behind to seek their fortune in the hippie camp that is called liquid.