It appears that it is currently (as of May 2016) impossible to get a permission from the U.S. for a solo tourist visit to Cuba.
However, in various older questions, users have stated that you can easily avoid the strict US travel restrictions by flying into Cuba from another country (such as Mexico or Canada) and avoiding getting an entry stamp in your U.S. passport.
According to the Embassy of Cuba in USA website:
Tourist Visa (Tourist Card)
The Tourist Visa or Tourist Card is only for purposes of tourism to Cuba. It is valid for one single entrance into national territory for a 30-day trip and can be extended for an additional 30 days at the office in the hotel where one has accommodations or with the immigration authority.
To obtain this visa in person at the Consulate, these documents are needed:
-Plane Ticket with entry and return dates
-Payment of the Consular fee for this service
As far as I can tell, the Cubans don't seem have harsh tourist travel restrictions for U.S. citizens.
However, according to some very old information on Wikitravel,
While in the past Cuba did not stamp US passports, allowing US travelers to visit Cuba without detection from their government, Cuba recently started stamping US passports as a matter of policy.
In light of the recent diplomatic relations, have Cubans tightened loopholes for U.S. tourists traveling without the blessing of the U.S.? Does Cuban customs still exempt entry stamps for travelers with a U.S. passports?