The case flew from Rotorua and arrived in Blenheim on Thursday. They were tested upon arrival after developing a sore throat.
"The initial test, including a follow-up swab, returned a weak positive result."
The ministry said the case is now isolating awaiting the results of further testing.
The case is considered low risk, due to their likely late stage of infection.
The ministry said initial case interviews suggested the case was linked to the Te Awamutu cluster.
A small number of close contacts had been identified. They are now isolating and will be tested.
The ministry said any locations of interest will be added to its website.
"We are also asking Blenheim residents with symptoms - no matter how mild - to please get tested, even if they are vaccinated."
Vaccination rates in the Marlborough region have seen 90 percent of residents receive their first dose, with 78 percent fully immunised.
It is the first community case in the South Island in almost a year.
Today's full case numbers are due at 1pm.
There were 129 cases reported yesterday and 102 cases on Thursday.