The DX Mail service, which is mostly used by colleagues in Legal and Governance, so also known as Legal post, is due to be cancelled at the end of September.
This change has arisen due to a significant fall in demand now that the majority of colleagues are working from home as a result of the restrictions to combat the Coronavirus (Covid-19) pandemic.
Letters and parcels will be sent by Royal Mail, using special delivery or signed for postage, once the DX Mail service ends.
Head of Property and Facilities Allan Whyte said: “As we all adjust to the new working world brought about by the Coronavirus, there will be changes.
“Whilst I appreciate change can be unsettling, there are alternatives, and, as a council, we also must balance the books, so there is little point in continuing with a service if it is not financially viable.”