PC Jones, who was not present at the hearing, was alleged to have been involved in the use of Class A drugs and in doing so breached the Standards of Professional Behaviour.
She was arrested in November 2015 following an internal investigation into allegations of drug use. She tested positive for drugs and traces of drugs were found to be present at her address.
The chair of the panel, Chief Constable of Durham Constabulary, Mike Barton said the behaviour "discredits the police service and undermines public confidence in it" and therefore there was no alternative but dismissal.
Chief Superintendent Judi Heaton, head of Professional Standards Branch, said: "Random drug testing takes place across the force on a regular basis and to date, all who have been tested have returned negative samples showing that those who are making bad decisions and possibly ignoring the severity of taking illegal substances are very small in number."