Pronounced the Italian way it's rather beautiful, don't you think? It means song (I guess you know this already!) - from the Latin 'carmen'.
Hang on, though. You mean the name, right? As in that actor in CSI: New York? Cos there's a colour 'carmine' too (deep red), and that is pronounced 'Car-mine', I think. Hmmm - actually not sure about that one... [confused emoticon]
Actually, although the words are spelt the same, they come from different roots - Carmine (the name) from Latin 'carmen'; and carmine the colour is from Arabic (? I think...) for red, IIRC. [definitely needs to get out more emoticon]