Two others can see new patients - but only for a one-off course of treatment and for emergency appointments.
British Dental Association Wales said the problem lies with capped dental budgets for treatment.
Both Cardiff and Vale health board and the Welsh government said extra resources have been made available to improve patient access.
According to the health board's website, only one practice in Cyncoed and one dental surgery in Pentwyn have spaces for new NHS patients.
A surgery in Rhiwbina says it will see new patients for a one-off course of treatment, while another in Roath said it only had emergency appointments.
Richard Birkin, national director for BDA Wales, claimed NHS dentists' ability to take on new patients are limited by the "target-driven" contract they have with NHS commissioners.
"We believe a better way of delivering care to patients is to have a contract that is based on prevention, rather than targets, with an in-built flexibility to allow dentists to take on more new patients," he said.
"Work has been underway for several years to road-test this preventive approach. It's now time that Welsh government has prevention at the core of the dental contract."
David Oliver, the health board's dental and optometry primary care support manager, said it would be improving new patient access following Â£200,000 of additional funding from the Welsh government.
He added: "The majority of practices do however accept patients onto a waiting list and will select patients from this waiting list as places become available.
"This means that over 2,700 new patients were treated in the Cardiff and Vale UHB area in May 2015 alone with up to 4,400 new patients being treated in March 2015."
A Welsh government spokesman said the number of people accessing NHS dental services in the health board area had increased in recent years and "additional resources" had been made available to the health board.
"We are currently examining changes to the dental contract and have been piloting new arrangements, which build in greater equity for patients and contractors, while ensuring improvements are made around quality and prevention," the spokesman added.
Just two out of 48 general dental practices in Cardiff are accepting new NHS patients.