Whether or not a rapist has any automatic parental rights or is able to petition the courts for custody or visitation is a state and not a federal matter. While this may change in the future — as this is one area where legislation is often proposed — for now, Florida does allow for a rapists parental rights to be terminated at the request of the child's mother. Understanding what the law allows for and everything that may come up during the termination process is an important part of being prepared.
While some states have laws that only allow for the termination of a rapist's parental rights in part, Florida's guidelines allow for all rights to be terminated. There also does not have to be a conviction for the rape on record for the termination.
The burden of proof is just "clear and convincing evidence" that the rape occurred and the child resulted from that rape. This can be a bit of a gray area and is just one part of the process where an attorney can be of great help in helping you understand what you will need to provide the courts.
In some states, the rapist who has had his parental rights terminated to the child resulting from the rape may still be required to pay child support. However, this is not true in Florida. Once the parental rights are terminated, the other party is no longer considered financially responsible for the child. While many victims would happily forego any child support payments to be assured that they won't have to deal with or coparent with their rapist on a regular basis, it's something to be aware of.
Source: RAINN, "Termination of Rapists' Parental Rights Laws," accessed March 22, 2016