If the current directory is empty, you can do that with:
git clone git@github.com:me/name.git .
(Note the . at the end to specify the current directory.) Of course, this also creates the .git directory in your current folder, not just the source code from your project.
This optional [directory] parameter is documented in the git clone manual page, which points out that cloning into an existing directory is only allowed if that directory is empty.