If it's not too much in your way you could leave it tied to a cane, feed it. bit and wit for more shoots to grow from th base. Do you know what variety of rose it is? Not a dog rose!?
I'm not sure how old "quite old" is but it obviously had enough oomph left to produce one good long shoot. Plus, when it finds out that it's no longer buried inside a laurel bush (hate 'em) it may be jolted into life. Nothing to lose!