The H5N8 strain found in a wild duck at an estuary near the Wildfowl and Wetland Trust (WWT) Llanelli Wetland Centre is the same which hit a turkey farm in Lincolnshire last week.
The centre said it had closed "as a precautionary measure".
Rural Affairs Secretary Lesley Griffiths said the risk to humans was "very low" and poultry was safe to eat.
The Welsh Government said it was the first time the H5N8 strain had been found in a wild bird in the UK.
It was described as "extremely concerning" by the Welsh Conservatives which called on the government to ensure safety measures were in place.
Restrictions were imposed across Britain to keep birds indoors after the disease came to light across Europe, the Middle East and north Africa.
On Tuesday, the measures were tightened to ban any indoor gatherings of birds at events such as livestock fairs, auctions and bird shows.
Ms Griffiths said: "This finding is not unexpected and follows calls for bird keepers to be more vigilant for signs of the disease. It is likely that more cases will be confirmed.
"There have been no reports of human infection from the H5N8 strain and the risk to human health is very low.
"The Food Standards Agency has also confirmed it is safe to eat poultry meat, such as turkey, goose and chicken."
In a statement on Thursday confirming its temporary closure, WWT Llanelli Wetland Centre said: "This is a concern for local birds - not humans.
"This is not a strain of bird flu that has ever transferred to people.
"The finding is not unexpected as the disease has already been found across Europe.
"Like other organisations across the country, we have been keeping a close eye out for any signs of the disease in birds this winter.
"We will remain closed while we increase surveillance among wild birds on our reserve and our collection of zoo birds, and put in place any measures necessary to help protect them."
Welsh Conservative spokesman for rural affairs, Paul Davies AM, said: "If it is allowed to spread then it could have a devastating impact on poultry farming across Wales.
"The Welsh Government must ensure that all farms provide details of their flocks to the poultry register and are appraised of all relevant safety measures in the event of an outbreak."
Chief Veterinary Officer Christianne Glossop urged people to report any sightings of dead waterfowl, or groups of at least five dead wild birds of other species in the same location, to the Animal and Plant Health Agency.

Strong winds helped the blaze to spread across the 336.1m (1,105ft) high building in the marina district.
One eyewitness told the BBC "there was debris flying everywhere, falling into neighbouring buildings". The fire was later extinguished.
It is not known what caused the blaze. Hundreds of people were evacuated and there are no reports of any casualties.
Footage posted on social media showed what looked like molten glass and pieces of masonry falling to the ground.
"In an hour-and-a-half the fire is almost out on about 60 floors, it looks like mostly the outer apartments that are damaged," eyewitness Rola told the BBC.
She said it looked like the blaze had started on the 50th floor.
"It's an iconic building, people were out on balconies all around it watching - even though it was 3am [23:00 GMT Saturday]," she said.
"The civil defence came really quickly and they have said there were no casualties, everybody was told to evacuate in a timely manner and we think everybody is OK."
One of the tower's residents, Kathryn Dickie, said she had seen panels from the building tumble to the ground.
"Those panels were catching fire and then coming off the building and being picked up by the wind," she said.
"And I was watching them, they were falling down on to the street still in flames, they weren't going out and they were actually knocking into the buildings across the road."
US resident RJ Morlock told the Associated Press news agency: "I was really surprised they got it under control pretty quickly. It looked like it was going to go up."
In 2012, a fire swept through Dubai's 34-storey Tamweel Tower.
No-one was injured in that incident but hundreds of residents were left homeless after the blaze, started reportedly by a cigarette butt.
The first finding of an infectious strain of avian flu in a UK wild bird has been confirmed in Carmarthenshire.
[NEXT_CONCEPT]
A fire has swept through the Torch skyscraper in Dubai, one of the tallest residential buildings in the world.