It will come into effect at 17:00 local time on Thursday (07:00 GMT).
The lockdown was called for the whole of the Australian Capital Territory - home to around 400,000 people - because authorities do not know how the infected person caught the virus.
Residents will only be able to leave home for essential reasons.
Queues were reported at supermarkets as the lockdown approached.
Australia is struggling to get on top of the highly infectious Delta strain, which has resulted in two of its largest cities - Sydney and Melbourne - going into strict lockdowns.
Large parts of New South Wales are locked down, and in Victoria there are similar restrictions on Melbourne for at least another week.
Extra military reinforcements could be called upon to ensure compliance with lockdown rules in Sydney, the New South Wales state government said.
"We are making sure that we do not leave any stone unturned in relation to extra (military) resources," state Premier Gladys Berejiklian told a media conference.
Some 580 unarmed army personnel are already assisting police in enforcing quarantine orders in areas of Sydney.
Despite seven weeks of lockdown in Sydney, infections remain near their highest. A total of 345 new locally acquired cases were recorded in New South Wales on Thursday, while two men in their 90s died.
Fewer than a quarter of Australians are vaccinated. Despite avoiding the worst of the pandemic, frustration over restrictions has led to several large anti-lockdown protests.