gloves, eye protection, and a surgical paper mask are a must here.
Then you are correct a good sharp scalpel - x-acto knife and cut down at the very base even taking some of the rock if you can. Rock is usually pretty soft and can be dug out with a knife like this. Then you have a rock pc. that you can super glue somewhere else.
Did I mention gloves, eye protection and a paper surgical mask as a must?