Mr Gutwein says by then he expects everyone in the state over the age of 12 will have had the opportunity to be vaccinated.
"On the 15th of December we will reopen our border," he said on Friday.
"We are not going to turn back from that date. If you're not vaccinated, get it done and get it done as soon as you can."
Interstate arrivals will need to provide a negative COVID-19 test within 72 hours of travel.
More than 70 per cent of Tasmanians aged over 16 are fully vaccinated and more than 85 per cent have received one dose.
The state is currently closed to New South Wales, Victoria and the ACT, while high-risk premises are declared in Queensland and South Australia.
The state has had just three cases this year on the back of a hard border approach but has dodged several COVID-19 bullets.
An infected NSW man, who recently sparked a three-day lockdown in the south after escaping hotel quarantine and spending 18 hours in the community, did not spread the virus.
The state government is enforcing mandatory vaccines for healthcare workers on October 31 and says it will sack anyone who doesn't meet the deadline.
AAP with The Project