Create a new repo (let's call it private-repo) via the Github UI. Then:
git clone --bare https://github.com/exampleuser/public-repo.git
cd public-repo.git
git push --mirror https://github.com/yourname/private-repo.git
cd ..
rm -rf public-repo.git
Reference: https://stackoverflow.com/questions/10065526/github-how-to-make-a-fork-of-public-repository-private