Brown said those people should not have to wait until last to receive their vaccinations, although more far-flung areas did pose a challenge.
The vaccine must be stored at an ultra low temperature, and southern stocks are being held in a special freezer in Christchurch.
Brown said the vaccine was being shipped to an SDHB facility, before being taken to the new clinic to be dispensed.
SDHB chief executive Chris Fleming praised staff who had worked on setting up the clinic, and said it was a remarkable effort that it had been set up in such a short time.
"Having faced the challenge and disruptions caused by Covid-19 over the past year, it is a great relief to now have the opportunity to be rolling out a vaccine to protect our population."
Around 575,000 vaccinations are required for everyone aged 16+ in the south to receive the required two doses, three weeks apart.