Public Health Wales (PHW) said 21 of the 35 residents at the unidentified home had been ill since 10 April.
It is unclear if the flu-like illness caused the death.
Earlier this week, it was revealed that two people had died after a flu outbreak at two Swansea nursing homes.
Public health officials said five residents had been hospitalised and the remainder had been offered the anti-viral drug, Tamiflu.
Carmarthenshire council, which runs the home, said it was working very closely with PHW.
Dr Mac Walapu, consultant in communicable disease control for PHW, said: "In view of the rapid spread and high attack rate of this illness, it is most likely to be influenza.
"We have also recently had outbreaks of influenza in nursing homes in Swansea.
"Flu circulates in the UK between October and April and is not uncommon to see outbreaks in closed communities like nursing homes, where the disease can spread easily."
Dr Walapu said people with flu would usually experience a fever, chills, headache, body aches and fatigue, and fit and healthy people would usually recover without any complications and without needing medication.
"However, in elderly people flu can be more serious and therefore we have treated residents at this home with Tamiflu as a precaution," he added.
Sheila Porter, Carmarthenshire council's head of primary, community and social care, said they were working very closely with PHW to support residents, visitors and staff by providing information, advice and reassurance.
"Sadly, one of our residents has passed away and although the death is not confirmed as linked to the flu-like illness, we extend our deepest sympathies to the family who we will continue to support," she added.
Earlier this week, PHW said two people had died and five others were in hospital after a flu outbreak at two Swansea nursing homes.
A total of 17 people have tested positive for the influenza.
Overall, 36 people, including six members of staff at the homes, have reported flu-like symptoms.
Health chiefs are investigating an outbreak of a flu-like illness following the death of a resident at a nursing home in Carmarthenshire.