There are flights between Cuba and Miami, but they are charter flights (often declared as humanitarian flights). Some evidence of this can be found here.

However, these flights cannot be booked as stand-alone flights, they are only available to Cuban expatriates living in the US and a very select group of other people.

As other posters have said, your best choice would be to fly via Nassau, Cancun or Jamaica (either Montego Bay or Kingston). You could also fly via Canada, but that will probably be more expensive.

One idea I would consider is to buy a roundtrip ticket Cancun-Havana-Cancun and then proceed to the States from CUN. It may work out to be cheaper than the two one-ways you have in mind.

One other point of advice - have you considered buying a round-the-world ticket? It's usually the cheapest way to string a series of one-way tickets together.