There is a deleted answer on this question that had a useful link: https://help.github.com/articles/duplicating-a-repository
The gist is
0. create the new empty repository (say, on github)
1. make a bare clone of the repository in some temporary location
2. change to the temporary location
3. perform a mirror-push to the new repository
4. change to another location and delete the temporary location
OP's example:
On your local machine
$ cd $HOME
$ git clone --bare https://git.fedorahosted.org/the/path/to/my_repo.git
$ cd my_repo.git
$ git push --mirror https://github.com/my_username/my_repo.git
$ cd ..
$ rm -rf my_repo.git