The remaining birds at the premises, near Gretna, will be humanely culled and a 3km Protection Zone and 10km Surveillance Zone have been declared around the infected premises to limit the risk of spread of the disease.
Within the zones, a range of different controls are now in place. These include restrictions on the movement of poultry, carcasses, eggs, used poultry litter and manure - and restrictions on bird gatherings.
Producers and bird keepers are being reminded to comply with the order to house birds.
Meanwhile, the chief veterinary officer for Wales, Christianne Glossop, confirmed the presence of avian flu on a mixed poultry premises near Crickhowell, Brecon and Radnorshire, Powys.
And a second suspected case has been identified in Northern Ireland, in a flock in Co Antrim.
Scotland's chief veterinary officer Sheila Voas said: "Keepers who are concerned about the health or welfare of their flock should seek veterinary advice immediately.
"Your private vet, or your local Animal and Plant Health Agency office, will also be able to provide practical advice on keeping your birds safe from infection.
"Any dead wild swans, geese, ducks or birds of prey, or five or more dead wild birds of other species (including gulls) in the same location at the same time, should be reported to Defra's national telephone helpline.
"Public health advice is that the risk to human health from the virus is very low and food standards bodies advise that avian influenzas pose a very low food safety risk for UK consumers, and it does not affect the consumption of poultry products including eggs."
Rural Affairs Secretary Mairi Gougeon said: "With the recent disease confirmations in wild and captive birds across the UK, it is not unexpected for avian influenza to be found in birds here in Scotland.
"We ask that the public remain vigilant and report any findings of dead wild birds to Defra's national telephone helpline.
"Do not touch or pick up any dead or sick birds that you find."