Unfortunately, yes. You're lucky, that's a relatively inexpensive fix. I'd recommend an Xacto knife with a new blade to cut the surround from the cone. A damp paper towel may dilute the glue enough to remove, or you may need to use some solvent (which in theory shouldn't harm a paper cone, but it may stain it, so test it on a hidden spot first), depending on what sort of glue was used.
Remember that when replacing the surround, you'll want to center it using a tone (many people recommend 30hz). Don't try to do it by feel (which is probably what the previous owner did).