For Java 1.6 and lower, I believe the safest and cleanest API for this is Guava's Files.move.
Example:
File newFile = new File(oldFile.getParent(), "new-file-name.txt");
Files.move(oldFile.toPath(), newFile.toPath());
The first line makes sure that the location of the new file is the same directory, i.e. the parent directory of the old file.
EDIT: I wrote this before I started using Java 7, which introduced a very similar approach. So if you're using Java 7+, you should see and upvote kr37's answer.