Document the behavior. This is done in order to have proof and records of the disruptive behavior. Collect emails from complaining employees, complaints from customers and any other tool you can use in order to monitor this employee's behavior.
Compare the behaviors you have documented and reviewed with the company's handbook on conduct. Most likely your company will have a list of unacceptable behaviors. Make the connection between the rude, arrogant and mean behavior you are witnessing with the handbook and document it, so you can bring this concern to the proper department.
Contact your company's human resources department. Set up a meeting to discuss what your options are with this subordinate. Prepare a file on the employee to show the human Resources person how this person is violating company policy and ask to set up a disciplinary meeting.
Distribute discipline based on your company's policy and human resources' suggestions. Many company and state laws prohibit firing for a first offense. In some cases you will have to issue a warning and discuss the problems with the employee. Provide the same proof you did this with HR when discussing with the employee.