Travellers must have received their second vaccination at least 14 days before departure and have evidence of a negative Covid-19 test before departure to be eligible for quarantine-free travel.
Despite the restrictions, the first positive Omicron case was detected in the country four days ago.
The Cook Islands Ministry of Health said the fourth person tested positive on Thursday after travelling to Aitutaki, an island in the country's north.
The person is in isolation and will isolate for at least the next 10 days.
Twenty-two other contacts took PCR tests but all returned negative results, the ministry said.
The Cook Islands has a double-dosed vaccination rate of 96 per cent of the eligible population, and 70 per cent have also had a booster shot.
Brown earlier said contact tracing teams had been working to control the outbreak.
"First and foremost, our incredibly high vaccination rate – one of the best in the world – is protecting all of us right now by slowing down the transmission of this virus from person to person and in many cases halting it completely," he said.
"Furthermore, for those vaccinated people who do contract Covid-19, it has been proven that the health impacts are reduced significantly.
"So, we can all rest a lot easier knowing that 96 per cent of our eligible population 12 years and older have been fully vaccinated and 99 per cent partially vaccinated – an outstanding achievement for our entire nation."
Brown said the country's health and safety protocols were doing their job and he was "extremely pleased" with the work government and non-government groups were putting in to restrict the spread of the virus.
The Cook Islands last month elected to continue with quarantine-free travel with New Zealand after Omicron was detected here.
