The Department for Environment, Food and Rural Affairs (Defra) said the risk to public health was very low. Some 6,000 birds will be culled and a 10km (6 mile) exclusion zone is in place.
The exact strain has not been confirmed but the H5N1 form, deadly to humans, has been ruled out by Defra officials.
The case is the first in the UK since 2008, when chickens on a farm in Banbury, Oxfordshire tested positive for the virus.
On Sunday, an outbreak of a highly contagious strain of bird flu was discovered at a poultry farm in the Netherlands.
The Dutch government has imposed a three-day nationwide ban on the transportation of poultry and eggs.
Officials say the strain, H5N8, is very dangerous for bird life and could potentially affect humans, although people can only be infected through very close contact with the affected birds.
UK chief veterinary officer Nigel Gibbens told the BBC the British case may be linked to the European outbreaks.
"The link to the disease they found in Germany and the Netherlands is our most likely source and, on that basis, Public Health England has said with this strain there is not a risk to public health," he said.
The exclusion zone around the farm in the village of Nafferton prevents all poultry and poultry waste being transferred in or out of the area.
Defra said the flu strain had been identified as the H5 virus, stressing that it was not the H5N1 strain.
A spokeswoman said: "We have confirmed a case of avian flu on a duck breeding farm in Yorkshire - the public health risk is very low and there is no risk to the food chain.
"We are taking immediate and robust action which includes introducing a restriction zone and culling all poultry on the farm to prevent any potential spread of infection. A detailed investigation is ongoing.
"We have a strong track record of controlling and eliminating previous outbreaks of avian flu in the UK."
A Public Health England spokesman said: "Public Health England are assisting Defra in the investigation of an avian flu outbreak at a duck breeding farm in Yorkshire.
"Based on what we know about this specific strain of avian influenza, the risk to human health in this case is considered extremely low."
by Michelle Roberts, health editor, BBC news website
There are many types of bird flu, and most are harmless to humans. Some strains, like H5N1 and H7N9, can potentially spread from birds to people if there is prolonged close contact.
Even then, the virus is unlikely to spread from person to person and there is no evidence that cooked poultry can infect people.
The ultimate concern is that a bird flu virus will one day mutate and acquire the ability to spread in people.
The most infamous pandemic - the 1918 Spanish flu, which killed millions - probably came from birds.
The UK's chief veterinary officer told the BBC it was a serious disease of poultry and it would be a few days before the exact strain would be known.
Mr Gibbens said the disease can be spread when poultry are moved, through the cages used to transport them and the people who work with the birds.
"It can also be spread by the wild bird population and that happens quite regularly," he said.
"There's a number of routes and we are looking to do as much as we can to find out the source and eliminate any further risks."
The farm's bio-security was good, he added, and they were checking other poultry farms within the exclusion zone. But he warned that more cases could follow.
"Because there's a wild bird risk we need farmers and their vets all over the country to be alert to possible disease in their farm that they can't explain, draw those to our attention so we can investigate quickly," he added.
According to the NHS, most types of bird flu are harmless to humans.
But two strains, H5N1 and H7N9, have caused "serious concern in recent years".
H5N1 has a mortality rate of about 60% within infected humans, according to the World Health Organisation. The UN's public health arm recorded 377 deaths from H5N1 in 15 countries by July last year.
H7N9 was first detected in China in 2013 and has led to dozens of deaths.