Found a command line solution of the same length as Oliver Refalo's, using good ol' awk:
git branch | awk '/^\*/{print $2}'
awk
reads that as "do the stuff in {}
on lines matching the regex". By default it assumes whitespace-delimited fields, so you print the second. If you can assume that only the line with your branch has the *, you can drop the ^. Ah, bash golf!