Avi's solution is probably easiest, but has the downside that the new repository will contain the full history of all the stuff outside of this one project. In the example below, I'm assuming that you are at the command line of a *NIX machine and that you are in a directory containing your existing repository.
svnadmin create new-project-repo # create new repo next to old repo
svnadmin dump old-project-repo | svnadmin load new-project-repo
svn mv file://$PWD/new-project-repo/my-project/trunk file://$PWD/new-project-repo/trunk
svn mv file://$PWD/new-project-repo/my-project/branches file://$PWD/new-project-repo/branches
svn mv file://$PWD/new-project-repo/my-project/tags file://$PWD/new-project-repo/tags
# ... svn rm file://$PWD/new-project-repo/XXX for all the cruft still at the top level.
If you are running subversion 1.6.x, svnsync provides a better option. Prior to 1.6.x, svnsync could only be used to copy whole repositories. With 1.6.x it has gained the ability to copy only a given subtree of the repository.
svnadmin create new-project-repo
# svnsync needs the ability to set revision properties, so we must install a pre-revprop-change
# hook script which just exits with status 0 (no error). On *NIX systems, the program `true` will do this.
ln -s /usr/bin/true new-project-repo/hooks/pre-revprop-change # possibly /bin/true
svnsync init file://$PWD/new-project-repo file://$PWD/old-project-repo/my-project
svnsync sync file://$PWD/new-project-repo
# go drink some coffee