The stall members at the Bathgate Delivery Office have been sent home.
Royal Mail said that post is still being delivered as usual and the office remains open. Some local residents reported receiving mail on Sunday this weekend because of disruption caused to the centre.
The building on Inchmuir Road has been thoroughly cleaned since the positive tests were confirmed, Royal Mail said.
It comes after one Royal Mail staff member tested positive for the virus at a Fife delivery office in September.
Three fellow workers at the Dunfermline office on Woodmill Street were told to self-isolate after having come into contact with the colleague who tested positive.
A spokesperson for Royal Mail said the company had taken steps to ensure the safety of workers during the Covid-19 pandemic.
They said: “Royal Mail takes the health and safety of its colleagues, its customers and the local communities in which we operate very seriously. Ten of our colleagues recently tested positive for coronavirus at the Bathgate Delivery Office.
"They are now recuperating at home and we wish them a speedy recovery. We have carried out an intensive clean of the building. We are currently processing mail as normal at our Bathgate office. The office is open as normal.
“Throughout this crisis, every decision we make puts the health of our people and customers first. We have put in place a range of preventive measures to protect both our customers and our colleagues. We were the first UK company to put in place social-distancing measures in relation to parcel delivery."
The spokesperson added: "We pioneered contact-free delivery. We are temporarily not handing over our hand-held devices to customers to capture signatures. As well as encouraging good hand hygiene, standard ways of working have been revised to ensure that colleagues stay two metres apart at all times.
"In addition we have moved to a one person per van delivery model. All staff have been briefed about the social distancing measures jointly agreed by local management and the CWU. This has been supplemented with visible reminders such as posters and one-way floor markings.”