The Corams Lane facility may shut its doors in three weeks if cost-cutting proposals by Somerset County Council are pushed through.
Bosses say the move is an “integral part of changes” as it looks to find £100million in savings over the next four years.
In a letter to Wellington Town Council, the authority said: “When we considered all the information we concluded that while premises costs are minimal at Corams Lane, the low availability and take-up of appointments is not making best use of staff time.
“Making these changes is crucial in order to reshape the registration service across the county to be more efficient and to deliver a better customer service.”
The office is open eight hours a week over two days and is run by part-time staff.
The authority says no jobs will be lost and, subject to engagement, workers will be redeployed to the registration office at Taunton’s Old Municipal Building.
It also said talks are underway with Somerset Skills and Learning over the lease of the building.
Cllr Andrew Govier said he plans to write to the county council to see if it could open the facility once a week. He said: “It is a real shame that this facility is being lost, however I do accept that changes and savings need to be made.”
Cllr Janet Lloyd added: “I think it’s quite sad that the county council is proposing to close the office.
“When people have to register a death they are distraught enough anyway and when I lost somebody it was nice to be able to do that in Wellington.”
According to the county council, Wellington’s registration office is the least used out of 12 in Somerset.
Between April 2012 and March 2013, it said 156 births, 102 deaths and 34 notices of marriage or civil partnership were registered.
The facility will be closed subject to a county councillor decision.
The nearest registration office is at the Old Municipal Building, Taunton, which is open Monday to Friday 9am to 5.30pm.